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Hora (dance)

Hora, also known as horo and oro, is a type of circle dance originating in Aromanian and Romanian communities, especially in Romania and Moldova. It is also found in other South East European countries (such as Bulgaria and North Macedonia) and culturally adopted by ethnic minorities such as the Ashkenazi Jews (Yiddish: האָרע hore) and the Roma.

Etymology
The name, spelled differently in various countries, is derived from the Greek χορός (khorós): "dance", which is cognate with the Ancient Greek art form of χορεία (khoreía). The original meaning of the Greek word χορός may have been "circle".

Also, the words hora and oro are found in many Slavic languages and have the meaning of "round (dance)"; the verb oriti means "to speak, sound, sing" and previously meant "to celebrate".

The Greek χορός (khorós) is cognate with Pontic Greek χορόν (khorón), and has also given rise to the names of Bulgarian хоро (horo), Macedonian оро (oro), Romanian horă, kolo / коло in Serbian, the Turkish form hora and in Hebrew הורה (horah). The Khorumi dance of Georgia also might be connected to the Horon dance in the neighboring Turkish regions, as it rose out of the Adjara region, where Kartvelian Laz people co-existed for centuries with Greek Pontians. Overall, Horos has its origins in ancient Greece and Greek culture and tradition, it has spread to areas heavily influenced by the Greek culture and were part of the Byzantine Empire, these areas include most of the Balkans, Turkey, and Middle East.

Romania and Moldova
Horă (plural: hore) is a traditional Romanian and Moldovan folk dance where the dancers hold each other's hands and the circle spins, usually counterclockwise, as each participant follows a sequence of three steps forward and one step back. The dance is usually accompanied by musical instruments such as the cymbalom, accordion, violin, viola, double bass, saxophone, trumpet or the pan pipes.

The hora is popular during wedding celebrations and festivals, and is an essential part of the social entertainment in rural areas. One of the most famous hore is the "Hora Unirii" ("Hora of the Union"), which became a Romanian patriotic song as a result of being the hymn when Wallachia and Moldavia united to form the Principality of Romania in 1859. During the 2006/2007 New Year's Eve celebration, when Romania and Bulgaria joined the European Union, people were dancing Hora Bucuriei ("Hora of Joy") over the boulevards of Bucharest as a tribute to the EU anthem, Ode to Joy (Odă bucuriei). Some of the biggest hora circles can be found on early 20th century movies filmed by the Manaki brothers in Pindus, Greece, and performed by local Aromanians.

Horo in Bulgaria

The traditional Bulgarian dance horo (хоро) comes in many shapes. It is not necessary to be in a circle; a curving line of people is also acceptable. The steps used in a horo dance are extremely diverse. The horo may vary between three and seven or eight steps forward and one to five or six steps back, depending on the specific type.

Bulgarians believe that each village has their own type of horo. They differ by the rhythm of the music and the steps taken. There are no two horo dances with similar steps. There are probably over one hundred types of horo dances in the Bulgarian folklore.

In the past, the horo dance had a social role in Bulgarian society. It was mainly for fun, as a contest of skills, or for show, leading to the development of the variety of horo dances. There are hora for people with little skill that can be learned in five to ten minutes, but there are also very sophisticated dances that cannot be learned unless one is fluent in many of the simpler dances.

Oro in North Macedonia

North Macedonia uses the Cyrillic spelling of oro. The origins of the word oro vary from its use in socializing and celebrating to historical dancing before going into battle. Teshkoto, translated as "The difficult one", is one of those, danced by men only, the music of which reflects the sorrow and mood of war. The oro is danced in a circle, with men and women holding one another by hand. They are used to celebrating occasions such as weddings, christenings, name-days, national and religious holidays, graduations, and birthdays.

Roma horo
The horo is also popular among the Roma of Southeastern Europe, and the dancing is practically the same as that of the neighboring ethnicities. Roma Horos, and Roma music in general, are very much appreciated among the non-Roma in the Balkans, as they also have a reputation as skillful performers of other people's folk music.

Klezmer horah
In klezmer music, the horah refers to a circle dance. The horah has a slow, limping gait in triple meter, often three/eight time (3/8 time), and generally leads into a faster and more upbeat duple meter, usually a freylekh or a bulgar. Among Yiddish-speaking Jews, the triple-meter horah has also been called zhok (Romanian joc, 'dance') or krumer tants (Yiddish: 'crooked dance').

Israeli horah
The horah (הורה), which differs somewhat from that of some of the Eastern European countries, is widespread in the Jewish diaspora and played a foundational role in modern Israeli folk dancing. It became the symbol of the reconstruction of the country by the socialistic-agricultural Zionist movement. Although considered traditional, some claim it rose to popularity due to Hora Agadati, named after dancer and choreographer Baruch Agadati and performed for the first time in 1924. According to Gurit Kadman, the original melody was a Moldavian folk tune, which in mid-1940s was recognized by composer Uriya Boskovitz as an anti-Semitic one, and Gurit asked Boskovitz to write a new one. About the same time Ze’ev Havatselet wrote a lyrics to the tune (found, e.g., in the Library of Congress ). Now the dance is usually performed to Israeli folk songs, and sometimes to Jewish songs, often to the music of "Hava Nagila".

To start the dance, everybody forms a circle, holding hands or interlocking arms behind their backs or on their shoulders, and steps forward toward the left with the right foot, then follows with the left foot. The right foot is then brought back, followed by the left foot. This is done while holding hands and circling together in a fast and cheerful motion to the left. Large groups allow for the creation of several concentric circles, or a large spiral formation.

In the early days, horah was popular mainly in kibbutzim and small communities, often continuing for hours.

The horah became popular in group dances throughout Israel, and at weddings and other celebrations by Jews in Israel, the United States, United Kingdom, and Canada. The dance appeared in North America in the early 20th century, well before modern Israeli independence, brought directly from Eastern Europe by Jewish immigrants.

At Jewish weddings during the horah it is customary to raise the bride and groom, each on their own chair and holding a handkerchief between them, following Jewish tradition. This is also done at b'nai mitzvah, where the honoree and sometimes his or her family members are also raised on a chair, copying the wedding tradition.

The song "Hora", sung by Avi Toledano, who represented Israel in the Eurovision Song Contest 1982, is based on this dance.

Horon in Turkey
Horon in several variants is danced in Black Sea Region / Pontos of modern-day Turkey.

Oro in Montenegro
The oro (оро) circle dance should not be confused with the Montenegrin Oro dance of Montenegro and Herzegovina, which is a paired courtship dance. Its name comes from the Serbian оrао, meaning "eagle".

Perinița
Perinița is a traditional Romanian wedding folk dance. The dance is typical in the Muntenia region.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Hora (dance)


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Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (occasionally also called arcus functions,    antitrigonometric functions or cyclometric functions   ) are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

Notation

Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: $arcsin(x)$, $arccos(x)$, $arctan(x)$, etc. (This convention is used throughout this article.) This notation arises from the following geometric relationships: when measuring in radians, an angle of $θ$ radians will correspond to an arc whose length is $rθ$, where $r$ is the radius of the circle. Thus in the unit circle, the cosine of x function is both the arc and the angle, because the arc of a circle of radius 1 is the same as the angle. Or, "the arc whose cosine is $x$" is the same as "the angle whose cosine is $x$", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians. In computer programming languages, the inverse trigonometric functions are often called by the abbreviated forms asin, acos, atan.

The notations $sin^{−1}(x)$, $cos^{−1}(x)$, $tan^{−1}(x)$, etc., as introduced by John Herschel in 1813, are often used as well in English-language sources, much more than the also established $sin^{[−1]}(x)$, $cos^{[−1]}(x)$, $tan^{[−1]}(x)$ – conventions consistent with the notation of an inverse function, that is useful (for example) to define the multivalued version of each inverse trigonometric function: $$\tan^{-1}(x) = \{\arctan(x) + \pi k \mid k \in \mathbb Z\} ~.$$ However, this might appear to conflict logically with the common semantics for expressions such as $sin^{2}(x)$ (although only $sin^{2} x$, without parentheses, is the really common use), which refer to numeric power rather than function composition, and therefore may result in confusion between notation for the reciprocal (multiplicative inverse) and inverse function.

The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name — for example, $(cos(x))^{−1} = sec(x)$. Nevertheless, certain authors advise against using it, since it is ambiguous. Another precarious convention used by a small number of authors is to use an uppercase first letter, along with a “$−1$” superscript: $Sin^{−1}(x)$, $Cos^{−1}(x)$, $Tan^{−1}(x)$, etc. Although it is intended to avoid confusion with the reciprocal, which should be represented by $sin^{−1}(x)$, $cos^{−1}(x)$, etc., or, better, by $sin^{−1} x$, $cos^{−1} x$, etc., it in turn creates yet another major source of ambiguity, especially since many popular high-level programming languages (e.g. Mathematica and MAGMA) use those very same capitalised representations for the standard trig functions, whereas others (Python, SymPy, NumPy, Matlab, MAPLE, etc.) use lower-case.

Hence, since 2009, the ISO 80000-2 standard has specified solely the "arc" prefix for the inverse functions.

Principal values
Since none of the six trigonometric functions are one-to-one, they must be restricted in order to have inverse functions. Therefore, the result ranges of the inverse functions are proper (i.e. strict) subsets of the domains of the original functions.

For example, using in the sense of multivalued functions, just as the square root function $$y = \sqrt{x}$$ could be defined from $$y^2 = x,$$ the function $$y = \arcsin(x)$$ is defined so that $$\sin(y) = x.$$ For a given real number $$x,$$ with $$-1 \leq x \leq 1,$$ there are multiple (in fact, countably infinitely many) numbers $$y$$ such that $$\sin(y) = x$$; for example, $$\sin(0) = 0,$$ but also $$\sin(\pi) = 0,$$ $$\sin(2 \pi) = 0,$$ etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each $$x$$ in the domain, the expression $$\arcsin(x)$$ will evaluate only to a single value, called its principal value. These properties apply to all the inverse trigonometric functions.

The principal inverses are listed in the following table.

Note: Some authors define the range of arcsecant to be $( 0 \leq y < \frac{\pi}{2} \text{ or } \pi \leq y < \frac{3 \pi}{2} )$, because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, $$\tan(\arcsec(x)) = \sqrt{x^2 - 1},$$ whereas with the range $( 0 \leq y < \frac{\pi}{2} \text{ or } \frac{\pi}{2} < y \leq \pi )$, we would have to write $$\tan(\arcsec(x)) = \pm \sqrt{x^2 - 1},$$ since tangent is nonnegative on $0 \leq y < \frac{\pi}{2},$ but nonpositive on $\frac{\pi}{2} < y \leq \pi.$  For a similar reason, the same authors define the range of arccosecant to be $( - \pi < y \leq - \frac{\pi}{2}$  or $0 < y \leq \frac{\pi}{2} ) .$

Domains
If $$x$$ is allowed to be a complex number, then the range of $$y$$ applies only to its real part.

Solutions to elementary trigonometric equations
Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of $$2 \pi:$$

 * Sine and cosecant begin their period at $2 \pi k-\frac{\pi}{2}$ (where $$k$$ is an integer), finish it at $2 \pi k+\frac{\pi}{2},$  and then reverse themselves over $2 \pi k+\frac{\pi}{2}$  to $2 \pi k+\frac{3\pi}{2}.$
 * Cosine and secant begin their period at $$2 \pi k,$$ finish it at $$2 \pi k+\pi.$$ and then reverse themselves over $$2 \pi k+\pi$$ to $$2 \pi k+2 \pi.$$
 * Tangent begins its period at $2 \pi k-\frac{\pi}{2},$  finishes it at $2 \pi k+\frac{\pi}{2},$  and then repeats it (forward) over $2 \pi k+\frac{\pi}{2}$  to $2 \pi k+\frac{3 \pi}{2}.$
 * Cotangent begins its period at $$2 \pi k, $$ finishes it at $$2 \pi k+\pi,$$ and then repeats it (forward) over $$2 \pi k+\pi$$ to $$2 \pi k+2 \pi.$$

This periodicity is reflected in the general inverses, where $$k$$ is some integer.

The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions. It is assumed that the given values $$\theta,$$ $$r,$$ $$s,$$ $$x,$$ and $$y$$ all lie within appropriate ranges so that the relevant expressions below are well-defined. Note that "for some $$k \in \Z$$" is just another way of saying "for some integer $$k.$$"

The symbol $$\,\iff\,$$ is logical equality and indicates that if the left hand side is true then so is the right hand side and, conversely, if the right hand side is true then so is the left hand side (see this footnote The expression "LHS $$\,\iff\,$$ RHS" indicates that (a) the left hand side (i.e. LHS) and right hand side (i.e. RHS) are  true, or else (b) the left hand side and right hand side are  false; there is  option (c) (e.g. it is  possible for the LHS statement to be true and also simultaneously for the RHS statement to be false), because otherwise "LHS $$\,\iff\,$$ RHS" would not have been written.

To clarify, suppose that it is written "LHS $$\,\iff\,$$ RHS" where LHS (which abbreviates left hand side) and RHS are both statements that can individually be either be true or false. For example, if $$\theta$$ and $$s$$ are some given and fixed numbers and if the following is written:

then LHS is the statement "$$\tan \theta = s$$". Depending on what specific values $$\theta$$ and $$s$$ have, this LHS statement can either be true or false. For instance, LHS is true if $$\theta = 0$$ and $$s = 0$$ (because in this case $$\tan \theta = \tan 0 = s$$) but LHS is false if $$\theta = 0$$ and $$s = 2$$ (because in this case $$\tan \theta = \tan 0 = s$$ which is not equal to $$s = 2$$); more generally, LHS is false if $$\theta = 0$$ and $$s \neq 0.$$ Similarly, RHS is the statement "$$\theta = \arctan(s)+\pi k$$ for some $$k \in \Z$$". The RHS statement can also either true or false (as before, whether the RHS statement is true or false depends on what specific values $$\theta$$ and $$s$$ have). The logical equality symbol $$\,\iff\,$$ means that (a) if the LHS statement is true then the RHS statement is also true, and moreover (b) if the LHS statement is false then the RHS statement is also  false. Similarly, $$\,\iff\,$$ means that (c) if the RHS statement is true then the LHS statement is also  true, and moreover (d) if the RHS statement is false then the LHS statement is also  false. for more details and an example illustrating this concept).

where the first four solutions can be written in expanded form as:

For example, if $$\cos \theta = -1$$ then $$\theta = \pi+2 \pi k = -\pi+2 \pi (1+k)$$ for some $$k \in \Z.$$ While if $$\sin \theta = \pm 1$$ then $\theta = \frac{\pi}{2}+\pi k =-\frac{\pi}{2}+\pi (k+1)$ for some $$k \in \Z,$$ where $$k$$ will be even if $$\sin \theta = 1$$ and it will be odd if $$\sin \theta = -1.$$ The equations $$\sec \theta = -1$$ and $$\csc \theta = \pm 1$$ have the same solutions as $$\cos \theta = -1$$ and $$\sin \theta = \pm 1,$$ respectively. In all equations above for those just solved (i.e. except for $$\sin$$/$$\csc \theta = \pm 1$$ and $$\cos$$/$$\sec \theta =-1$$), the integer $$k$$ in the solution's formula is uniquely determined by $$\theta$$ (for fixed $$r, s, x,$$ and $$y$$).

With the help of integer parity $$\operatorname{Parity}(h) = \begin{cases} 0 & \text{if } h \text{ is even } \\ 1 & \text{if } h \text{ is odd } \\ \end{cases}$$ it is possible to write a solution to $$\cos \theta = x$$ that doesn't involve the "plus or minus" $$\,\pm\,$$ symbol:
 * $$cos \; \theta = x \quad$$ if and only if $$\quad \theta = (-1)^h \arccos(x) + \pi h + \pi \operatorname{Parity}(h) \quad$$ for some $$h \in \Z.$$

And similarly for the secant function,
 * $$sec \; \theta = r \quad$$ if and only if $$\quad \theta = (-1)^h \arcsec(r) + \pi h + \pi \operatorname{Parity}(h) \quad$$ for some $$h \in \Z,$$

where $$\pi h + \pi \operatorname{Parity}(h)$$ equals $$\pi h$$ when the integer $$h$$ is even, and equals $$\pi h + \pi$$ when it's odd.

Detailed example and explanation of the "plus or minus" symbol $x = sin(y)$
The solutions to $$\cos \theta = x$$ and $$\sec \theta = x$$ involve the "plus or minus" symbol $$\,\pm,\,$$ whose meaning is now clarified. Only the solution to $$\cos \theta = x$$ will be discussed since the discussion for $$\sec \theta = x$$ is the same. We are given $$x$$ between $$-1 \leq x \leq 1$$ and we know that there is an angle $$\theta$$ in some interval that satisfies $$\cos \theta = x.$$ We want to find this $$\theta.$$ The table above indicates that the solution is $$\,\theta = \pm \arccos x+2 \pi k\, \quad \text{ for some }k \in \Z$$ which is a shorthand way of saying that (at least) one of the following statement is true:

As mentioned above, if $$\,\arccos x = \pi\,$$ (which by definition only happens when $$x = \cos \pi = -1$$) then both statements (1) and (2) hold, although with different values for the integer $$k$$: if $$K$$ is the integer from statement (1), meaning that $$\theta = \pi+2 \pi K$$ holds, then the integer $$k$$ for statement (2) is $$K+1$$ (because $$\theta = -\pi+2 \pi (1+K)$$). However, if $$x \neq -1$$ then the integer $$k$$ is unique and completely determined by $$\theta.$$ If $$\,\arccos x = 0\,$$ (which by definition only happens when $$x = \cos 0 = 1$$) then $$\,\pm\arccos x = 0\,$$ (because $$\,+ \arccos x = +0 = 0\,$$ and $$\,-\arccos x = -0 = 0\,$$ so in both cases $$\,\pm \arccos x\,$$ is equal to $$0$$) and so the statements (1) and (2) happen to be identical in this particular case (and so both hold). Having considered the cases $$\,\arccos x = 0\,$$ and $$\,\arccos x = \pi,\,$$ we now focus on the case where $$\,\arccos x \neq 0\,$$ and $$\,\arccos x \neq \pi,\,$$ So assume this from now on. The solution to $$\cos \theta = x$$ is still $$\,\theta = \pm \arccos x+2 \pi k\, \quad \text{ for some }k \in \Z$$ which as before is shorthand for saying that one of statements (1) and (2) is true. However this time, because $$\,\arccos x \neq 0\,$$ and $$\,0 < \arccos x < \pi,\,$$ statements (1) and (2) are different and furthermore, exactly one of the two equalities holds (not both). Additional information about $$\theta$$ is needed to determine which one holds. For example, suppose that $$x = 0$$ and that that is known about $$\theta$$ is that $$\,-\pi \leq \theta \leq \pi\,$$ (and nothing more is known). Then $$\arccos x = \arccos 0 = \frac{\pi}{2}$$ and moreover, in this particular case $$k = 0$$ (for both the $$\,+\,$$ case and the $$\,-\,$$ case) and so consequently, $$\theta ~=~ \pm \arccos x+2 \pi k ~=~ \pm \left(\frac{\pi}{2}\right)+2\pi (0) ~=~ \pm \frac{\pi}{2}.$$ This means that $$\theta$$ could be either $$\,\pi/2\,$$ or $$\,-\pi/2.$$ Without additional information it is not possible to determine which of these values $$\theta$$ has. An example of some additional information that could determine the value of $$\theta$$ would be knowing that the angle is above the $$x$$-axis (in which case $$\theta = \pi/2$$) or alternatively, knowing that it is below the $$x$$-axis (in which case $$\theta =-\pi/2$$).
 * $$\,\theta = \arccos x+2 \pi k\,$$ for some integer $$k,$$ or
 * $$\,\theta =-\arccos x+2 \pi k\,$$ for some integer $$k.$$

Equal identical trigonometric functions

 * Set of all solutions to elementary trigonometric equations

Thus given a single solution $$\theta$$ to an elementary trigonometric equation ($$\sin \theta = y$$ is such an equation, for instance, and because $$\sin (\arcsin y) = y$$ always holds, $$\theta := \arcsin y$$ is always a solution), the set of all solutions to it are:

Transforming equations
The equations above can be transformed by using the reflection and shift identities:

These formulas imply, in particular, that the following hold:

$$ \begin{align} \sin \theta &= -\sin(-\theta) &&= -\sin(\pi+\theta) &&= \phantom{-}\sin(\pi-\theta) \\ &= -\cos\left(\frac{\pi}{2}+\theta\right) &&= \phantom{-}\cos\left(\frac{\pi}{2}-\theta\right) &&= -\cos\left(-\frac{\pi}{2}-\theta\right) \\ &= \phantom{-}\cos\left(-\frac{\pi}{2}+\theta\right) &&= -\cos\left(\frac{3\pi}{2}-\theta\right) &&= -\cos\left(-\frac{3\pi}{2}+\theta\right) \\[0.3ex] \cos \theta &= \phantom{-}\cos(-\theta) &&= -\cos(\pi+\theta) &&= \phantom{-}\cos(\pi-\theta) \\ &= \phantom{-}\sin\left(\frac{\pi}{2}+\theta\right) &&= \phantom{-}\sin\left(\frac{\pi}{2}-\theta\right) &&= -\sin\left(-\frac{\pi}{2}-\theta\right) \\ &= -\sin\left(-\frac{\pi}{2}+\theta\right) &&= -\sin\left(\frac{3\pi}{2}-\theta\right) &&= \phantom{-}\sin\left(-\frac{3\pi}{2}+\theta\right) \\[0.3ex] \tan \theta &= -\tan(-\theta) &&= \phantom{-}\tan(\pi+\theta) &&= -\tan(\pi-\theta) \\ &= -\cot\left(\frac{\pi}{2}+\theta\right) &&= \phantom{-}\cot\left(\frac{\pi}{2}-\theta\right) &&= \phantom{-}\cot\left(-\frac{\pi}{2}-\theta\right) \\ &= -\cot\left(-\frac{\pi}{2}+\theta\right) &&= \phantom{-}\cot\left(\frac{3\pi}{2}-\theta\right) &&= -\cot\left(-\frac{3\pi}{2}+\theta\right) \\[0.3ex] \end{align} $$

where swapping $$\sin \leftrightarrow \csc,$$ swapping $$\cos \leftrightarrow \sec,$$ and swapping $$\tan \leftrightarrow \cot$$ gives the analogous equations for $$\csc, \sec, \text{ and } \cot,$$ respectively.

So for example, by using the equality $\sin \left(\frac{\pi}{2}-\theta\right) = \cos \theta,$ the equation $$\cos \theta = x$$ can be transformed into $\sin \left(\frac{\pi}{2}-\theta\right) = x,$  which allows for the solution to the equation $$\;\sin \varphi = x\;$$ (where $\varphi := \frac{\pi}{2}-\theta$ ) to be used; that solution being: $$\varphi = (-1)^k \arcsin (x)+\pi k \; \text{ for some } k \in \Z,$$ which becomes: $$\frac{\pi}{2}-\theta ~=~ (-1)^k \arcsin (x)+\pi k \quad \text{ for some } k \in \Z$$ where using the fact that $$(-1)^{k} = (-1)^{-k}$$ and substituting $$h :=-k$$ proves that another solution to $$\;\cos \theta = x\;$$ is: $$\theta ~=~ (-1)^{h+1} \arcsin (x)+\pi h+\frac{\pi}{2} \quad \text{ for some } h \in \Z.$$ The substitution $$\;\arcsin x = \frac{\pi}{2}-\arccos x\;$$ may be used express the right hand side of the above formula in terms of $$\;\arccos x\;$$ instead of $$\;\arcsin x.\;$$

Relationships between trigonometric functions and inverse trigonometric functions
Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of length $$x,$$ then applying the Pythagorean theorem and definitions of the trigonometric ratios. It is worth noting that for arcsecant and arccosecant, the diagram assumes that $$x$$ is positive, and thus the result has to be corrected through the use of absolute values and the signum (sgn) operation.

Relationships among the inverse trigonometric functions

Complementary angles:
 * $$\begin{align}

\arccos(x) &= \frac{\pi}{2} - \arcsin(x) \\[0.5em] \arccot(x) &= \frac{\pi}{2} - \arctan(x) \\[0.5em] \arccsc(x) &= \frac{\pi}{2} - \arcsec(x) \end{align}$$

Negative arguments:
 * $$\begin{align}

\arcsin(-x) &= -\arcsin(x) \\ \arccsc(-x) &= -\arccsc(x) \\ \arccos(-x) &= \pi -\arccos(x) \\ \arcsec(-x) &= \pi -\arcsec(x) \\ \arctan(-x) &= -\arctan(x) \\ \arccot(-x) &= \pi -\arccot(x) \end{align}$$

Reciprocal arguments:
 * $$\begin{align}

\arcsin\left(\frac{1}{x}\right) &= \arccsc(x) & \\[0.3em] \arccsc\left(\frac{1}{x}\right) &= \arcsin(x) & \\[0.3em] \arccos\left(\frac{1}{x}\right) &= \arcsec(x) & \\[0.3em] \arcsec\left(\frac{1}{x}\right) &= \arccos(x) & \\[0.3em] \arctan\left(\frac{1}{x}\right) &= \arccot(x)      &= \frac{\pi}{2} - \arctan(x) \,, \text{ if } x > 0 \\[0.3em] \arctan\left(\frac{1}{x}\right) &= \arccot(x) - \pi &= -\frac{\pi}{2} - \arctan(x) \,, \text{ if } x < 0 \\[0.3em] \arccot\left(\frac{1}{x}\right) &= \arctan(x)      &= \frac{\pi}{2} - \arccot(x) \,, \text{ if } x > 0 \\[0.3em] \arccot\left(\frac{1}{x}\right) &= \arctan(x) + \pi &= \frac{3\pi}{2} - \arccot(x) \,, \text{ if } x < 0 \end{align}$$ The identities above can be used with (and derived from) the fact that $$\sin$$ and $$\csc$$ are reciprocals (i.e. $$\csc = \tfrac1{\sin}$$), as are $$\cos$$ and $$\sec,$$ and $$\tan$$ and $$\cot.$$

Useful identities if one only has a fragment of a sine table:
 * $$\begin{align}

\arcsin(x) &= \frac{1}{2}\arccos\left(1-2x^2\right) \,, \text{ if } 0 \leq x \leq 1 \\ \arcsin(x) &= \arctan\left(\frac{x}{\sqrt{1 - x^2}}\right) \\ \arccos(x) &= \frac{1}{2}\arccos\left(2x^2-1\right) \,, \text{ if } 0 \leq x \leq 1 \\ \arccos(x) &= \arctan\left(\frac{\sqrt{1 - x^2}}{x}\right) \\ \arccos(x) &= \arcsin\left(\sqrt{1 - x^2}\right) \,, \text{ if } 0 \leq x \leq 1 \text{ , from which you get } \\ \arccos   &\left(\frac{1-x^2}{1 + x^2}\right) = \arcsin \left (\frac{2x}{1 + x^2}\right) \,, \text{ if } 0 \leq x \leq 1 \\ \arcsin   &\left(\sqrt{1 - x^2}\right) =\frac{\pi}{2}-\sgn(x)\arcsin(x) \\ \arctan(x) &= \arcsin\left(\frac{x}{\sqrt{1 + x^2}}\right) \\ \arccot(x) &= \arccos\left(\frac{x}{\sqrt{1 + x^2}}\right) \end{align}$$

Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real).

A useful form that follows directly from the table above is

 * $$\arctan(x) = \arccos\left(\sqrt{\frac{1}{1+x^2}}\right)\,, \text{ if } x\geq 0 $$.

It is obtained by recognizing that $$\cos\left(\arctan\left(x\right)\right) = \sqrt{\frac{1}{1+x^2}} = \cos\left(\arccos\left(\sqrt{\frac{1}{1+x^2}}\right)\right)$$.

From the half-angle formula, $$\tan\left(\tfrac{\theta}{2}\right) = \tfrac{\sin(\theta)}{1 + \cos(\theta)}$$, we get:
 * $$\begin{align}

\arcsin(x) &= 2 \arctan\left(\frac{x}{1 + \sqrt{1 - x^2}}\right) \\[0.5em] \arccos(x) &= 2 \arctan\left(\frac{\sqrt{1 - x^2}}{1 + x}\right) \,, \text{ if } -1 < x \leq 1 \\[0.5em] \arctan(x) &= 2 \arctan\left(\frac{x}{1 + \sqrt{1 + x^2}}\right) \end{align}$$

Arctangent addition formula

 * $$\arctan(u) \pm \arctan(v) = \arctan\left(\frac{u \pm v}{1 \mp uv}\right) \pmod \pi \,, \quad u v \ne 1 \, .$$

This is derived from the tangent addition formula
 * $$\tan(\alpha \pm \beta) = \frac{\tan(\alpha) \pm \tan(\beta)}{1 \mp \tan(\alpha)\tan(\beta)} \, ,$$

by letting
 * $$\alpha = \arctan(u) \,, \quad \beta = \arctan(v) \, .$$

Derivatives of inverse trigonometric functions
The derivatives for complex values of z are as follows:
 * $$\begin{align}

\frac{d}{dz} \arcsin(z) &{} = \frac{1}{\sqrt{1-z^2}} \; ;  &z &{}\neq -1, +1 \\ \frac{d}{dz} \arccos(z) &{} = -\frac{1}{\sqrt{1-z^2}} \; ; &z &{}\neq -1, +1 \\ \frac{d}{dz} \arctan(z) &{} = \frac{1}{1+z^2} \; ;         &z &{}\neq -i, +i\\ \frac{d}{dz} \arccot(z) &{} = -\frac{1}{1+z^2} \; ;        &z &{}\neq -i, +i \\ \frac{d}{dz} \arcsec(z) &{} = \frac{1}{z^2 \sqrt{1 - \frac{1}{z^{2}}}} \; ;  &z &{}\neq -1, 0, +1 \\ \frac{d}{dz} \arccsc(z) &{} = -\frac{1}{z^2 \sqrt{1 - \frac{1}{z^{2}}}} \; ; &z &{}\neq -1, 0, +1 \end{align}$$ Only for real values of x:
 * $$\begin{align}

\frac{d}{dx} \arcsec(x) &{} = \frac{1}{|x| \sqrt{x^2-1}} \; ; & |x| > 1\\ \frac{d}{dx} \arccsc(x) &{} = -\frac{1}{|x| \sqrt{x^2-1}} \; ; & |x| > 1 \end{align}$$

These formulas can be derived in terms of the derivatives of trigonometric functions. For example, if $$x = \sin \theta$$, then $dx/d\theta = \cos \theta = \sqrt{1-x^2},$ so
 * $$\frac{d}{dx}\arcsin(x) = \frac{d \theta}{dx} = \frac{1}{dx/d\theta} = \frac{1}{\sqrt{1-x^2}}.$$

Expression as definite integrals
Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral:
 * $$\begin{align}

\arcsin(x) &{}= \int_0^x \frac{1}{\sqrt{1 - z^2}} \, dz \;, & |x| &{} \leq 1\\ \arccos(x) &{}= \int_x^1 \frac{1}{\sqrt{1 - z^2}} \, dz \;, & |x| &{} \leq 1\\ \arctan(x) &{}= \int_0^x \frac{1}{z^2 + 1} \, dz \; ,\\ \arccot(x) &{}= \int_x^\infty \frac{1}{z^2 + 1} \, dz \; ,\\ \arcsec(x) &{}= \int_1^x \frac{1}{z \sqrt{z^2 - 1}} \, dz = \pi + \int_{-x}^{-1} \frac{1}{z \sqrt{z^2 - 1}} \, dz\;, & x &{} \geq 1\\ \arccsc(x) &{}= \int_x^\infty \frac{1}{z \sqrt{z^2 - 1}} \, dz = \int_{-\infty}^{-x} \frac{1}{z \sqrt{z^2 - 1}} \, dz \;, & x &{} \geq 1\\ \end{align}$$ When x equals 1, the integrals with limited domains are improper integrals, but still well-defined.

Infinite series
Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using power series, as follows. For arcsine, the series can be derived by expanding its derivative, $\tfrac{1}{\sqrt{1-z^2}}$, as a binomial series, and integrating term by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding its derivative $\frac{1}{1+z^2}$ in a geometric series, and applying the integral definition above (see Leibniz series).

\begin{align} \arcsin(z) & = z + \left( \frac{1}{2} \right) \frac{z^3}{3} + \left( \frac{1 \cdot 3}{2 \cdot 4} \right) \frac{z^5}{5} + \left( \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \right) \frac{z^7}{7} + \cdots \\[5pt] & = \sum_{n=0}^\infty \frac{(2n-1)!!}{(2n)!!}\frac{z^{2n+1}}{2n+1} \\[5pt] & = \sum_{n=0}^\infty \frac{(2n)!}{(2^n n!)^2} \frac{z^{2n+1}}{2n+1} \, ; \qquad |z| \le 1 \end{align} $$

 * $$\arctan(z)

= z - \frac{z^3}{3} +\frac{z^5}{5} - \frac{z^7}{7} + \cdots = \sum_{n=0}^\infty \frac{(-1)^n z^{2n+1}}{2n+1} \, ; \qquad |z| \le 1 \qquad z \neq i,-i$$

Series for the other inverse trigonometric functions can be given in terms of these according to the relationships given above. For example, $$\arccos(x) = \pi/2 - \arcsin(x)$$, $$\arccsc(x) = \arcsin(1/x)$$, and so on. Another series is given by:

 * $$2\left(\arcsin\left(\frac{x}{2}\right) \right)^2 = \sum_{n=1}^\infty \frac{x^{2n}}{n^2\binom {2n} n}.$$

Leonhard Euler found a series for the arctangent that converges more quickly than its Taylor series:

 * $$\arctan(z) = \frac z {1 + z^2} \sum_{n=0}^\infty \prod_{k=1}^n \frac{2k z^2}{(2k + 1)(1 + z^2)}.$$

(The term in the sum for n = 0 is the empty product, so is 1.)

Alternatively, this can be expressed as

 * $$\arctan(z) = \sum_{n=0}^\infty \frac{2^{2n} (n!)^2}{(2n + 1)!} \frac{z^{2n + 1}}{(1 + z^2)^{n + 1}}.$$

Another series for the arctangent function is given by

 * $$\arctan(z) = i\sum_{n=1}^\infty\frac{1}{2n - 1}\left(\frac{1}{(1 + 2i/z)^{2n-1}} - \frac{1}{(1 - 2i/z)^{2n - 1}}\right),$$

where $$i=\sqrt{-1}$$ is the imaginary unit.

Continued fractions for arctangent
Two alternatives to the power series for arctangent are these generalized continued fractions:

 * $$\arctan(z) =

\frac z {1 + \cfrac{(1z)^2}{3 - 1z^2 + \cfrac{(3z)^2}{5 - 3z^2 + \cfrac{(5z)^2}{7 - 5z^2 + \cfrac{(7z)^2}{9-7z^2 + \ddots}}}}} = \frac{z}{1 + \cfrac{(1z)^2}{3 + \cfrac{(2z)^2}{5 + \cfrac{(3z)^2}{7 + \cfrac{(4z)^2}{9 + \ddots}}}}} $$

The second of these is valid in the cut complex plane. There are two cuts, from −i to the point at infinity, going down the imaginary axis, and from i to the point at infinity, going up the same axis. It works best for real numbers running from −1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the first) are just (nz)2, with each perfect square appearing once. The first was developed by Leonhard Euler; the second by Carl Friedrich Gauss utilizing the Gaussian hypergeometric series.

Indefinite integrals of inverse trigonometric functions
For real and complex values of z:
 * $$\begin{align}

\int \arcsin(z) \, dz &{}= z \, \arcsin(z) + \sqrt{1 - z^2} + C\\ \int \arccos(z) \, dz &{}= z \, \arccos(z) - \sqrt{1 - z^2} + C\\ \int \arctan(z) \, dz &{}= z \, \arctan(z) - \frac{1}{2} \ln \left( 1 + z^2 \right) + C\\ \int \arccot(z) \, dz &{}= z \, \arccot(z) + \frac{1}{2} \ln \left( 1 + z^2 \right) + C\\ \int \arcsec(z) \, dz &{}= z \, \arcsec(z) - \ln \left[ z \left( 1 + \sqrt{ \frac{z^2-1}{z^2} } \right) \right] + C\\ \int \arccsc(z) \, dz &{}= z \, \arccsc(z) + \ln \left[ z \left( 1 + \sqrt{ \frac{z^2-1}{z^2} } \right) \right] + C \end{align}$$

For real x ≥ 1:
 * $$\begin{align}

\int \arcsec(x) \, dx &{}= x \, \arcsec(x) - \ln \left( x + \sqrt{x^2-1} \right) + C\\ \int \arccsc(x) \, dx &{}= x \, \arccsc(x) + \ln \left( x + \sqrt{x^2-1} \right) + C \end{align}$$

For all real x not between -1 and 1:
 * $$\begin{align}

\int \arcsec(x) \, dx &{}= x \, \arcsec(x) - \sgn(x) \ln\left| x + \sqrt{x^2-1}\right| + C\\ \int \arccsc(x) \, dx &{}= x \, \arccsc(x) + \sgn(x) \ln\left| x + \sqrt{x^2-1}\right| + C \end{align}$$

The absolute value is necessary to compensate for both negative and positive values of the arcsecant and arccosecant functions. The signum function is also necessary due to the absolute values in the derivatives of the two functions, which create two different solutions for positive and negative values of x. These can be further simplified using the logarithmic definitions of the inverse hyperbolic functions:
 * $$\begin{align}

\int \arcsec(x) \, dx &{}= x \, \arcsec(x) - \operatorname{arcosh}(|x|) + C\\ \int \arccsc(x) \, dx &{}= x \, \arccsc(x) + \operatorname{arcosh}(|x|) + C\\ \end{align}$$

The absolute value in the argument of the arcosh function creates a negative half of its graph, making it identical to the signum logarithmic function shown above.

All of these antiderivatives can be derived using integration by parts and the simple derivative forms shown above.

Example
Using $$\int u \, dv = u v - \int v \, du$$ (i.e. integration by parts), set

 * $$\begin{align}

u &= \arcsin(x) & dv &= dx \\ du &= \frac{dx}{\sqrt{1-x^2}} & v &= x \end{align}$$

Then

 * $$\int \arcsin(x) \, dx = x \arcsin(x) - \int \frac{x}{\sqrt{1-x^2}} \, dx,$$

which by the simple substitution $$w=1-x^2,\ dw = -2x\,dx$$ yields the final result:

 * $$\int \arcsin(x) \, dx = x \arcsin(x) + \sqrt{1-x^2} + C $$

Extension to the complex plane

Since the inverse trigonometric functions are analytic functions, they can be extended from the real line to the complex plane. This results in functions with multiple sheets and branch points. One possible way of defining the extension is:
 * $$\arctan(z) = \int_0^z \frac{dx}{1 + x^2} \quad z \neq -i, +i $$

where the part of the imaginary axis which does not lie strictly between the branch points (−i and +i) is the branch cut between the principal sheet and other sheets. The path of the integral must not cross a branch cut. For z not on a branch cut, a straight line path from 0 to z is such a path. For z on a branch cut, the path must approach from Re[x] > 0 for the upper branch cut and from Re[x] < 0 for the lower branch cut.

The arcsine function may then be defined as:

 * $$\arcsin(z) = \arctan\left(\frac{z}{\sqrt{1 - z^2}}\right) \quad z \neq -1, +1 $$

where (the square-root function has its cut along the negative real axis and) the part of the real axis which does not lie strictly between −1 and +1 is the branch cut between the principal sheet of arcsin and other sheets;
 * $$\arccos(z) = \frac{\pi}{2} - \arcsin(z) \quad z \neq -1, +1 $$

which has the same cut as arcsin;
 * $$\arccot(z) = \frac{\pi}{2} - \arctan(z) \quad z \neq -i, i $$

which has the same cut as arctan;
 * $$\arcsec(z) = \arccos\left(\frac{1}{z}\right) \quad z \neq -1, 0, +1 $$

where the part of the real axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and other sheets;
 * $$\arccsc(z) = \arcsin\left(\frac{1}{z}\right) \quad z \neq -1, 0, +1 $$

which has the same cut as arcsec.

Logarithmic forms
These functions may also be expressed using complex logarithms. This extends their domains to the complex plane in a natural fashion. The following identities for principal values of the functions hold everywhere that they are defined, even on their branch cuts.

 * $$\begin{align}

\arcsin(z) &{}= -i \ln \left( \sqrt{1-z^2} + iz \right) = i \ln \left( \sqrt{1-z^2} - iz \right) &{}= \arccsc\left(\frac{1}{z}\right) \\[10pt] \arccos(z) &{}= -i \ln \left( i \sqrt {1-z^2} + z \right) = \frac{\pi}{2} - \arcsin(z) &{}= \arcsec\left(\frac{1}{z}\right) \\[10pt] \arctan(z) &{}= -\frac{i}{2}\ln \left(\frac{i - z}{i + z}\right) = -\frac{i}{2}\ln \left(\frac{1 + iz}{1 - iz}\right) &{}= \arccot\left(\frac{1}{z}\right) \\[10pt] \arccot(z) &{}= -\frac{i}{2}\ln\left( \frac{z + i}{z - i} \right) = -\frac{i}{2}\ln\left( \frac{iz - 1}{iz + 1} \right) &{}= \arctan\left(\frac{1}{z}\right) \\[10pt] \arcsec(z) &{}= -i \ln \left( i \sqrt{1 - \frac{1}{z^2}} + \frac{1}{z} \right) = \frac{\pi}{2} - \arccsc(z) &{}= \arccos\left(\frac{1}{z}\right) \\[10pt] \arccsc(z) &{}= -i \ln \left( \sqrt{1 - \frac{1}{z^2}} + \frac{i}{z} \right) = i \ln \left( \sqrt{1 - \frac{1}{z^2}} - \frac{i}{z} \right) &{}= \arcsin\left(\frac{1}{z}\right) \end{align}$$

Generalization
Because all of the inverse trigonometric functions output an angle of a right triangle, they can be generalized by using Euler's formula to form a right triangle in the complex plane. Algebraically, this gives us:

 * $$ce^{i\theta} = c\cos(\theta) + ic\sin(\theta)$$

or
 * $$ce^{i\theta} = a + ib$$

where $$a$$ is the adjacent side, $$b$$ is the opposite side, and $$c$$ is the hypotenuse. From here, we can solve for $$\theta$$.

 * $$\begin{align}

e^{\ln(c) + i\theta} & = a + ib \\ \ln c + i\theta & = \ln(a + ib) \\ \theta & = \operatorname{Im}\left( \ln(a + ib) \right) \end{align}$$ or
 * $$\theta = -i\ln\left(\frac{a + ib}{c}\right)$$

Simply taking the imaginary part works for any real-valued $$a$$ and $$b$$, but if $$a$$ or $$b$$ is complex-valued, we have to use the final equation so that the real part of the result isn't excluded. Since the length of the hypotenuse doesn't change the angle, ignoring the real part of $$\ln(a+bi)$$ also removes $$c$$ from the equation. In the final equation, we see that the angle of the triangle in the complex plane can be found by inputting the lengths of each side. By setting one of the three sides equal to 1 and one of the remaining sides equal to our input $$z$$, we obtain a formula for one of the inverse trig functions, for a total of six equations. Because the inverse trig functions require only one input, we must put the final side of the triangle in terms of the other two using the Pythagorean Theorem relation
 * $$a^2 + b^2 = c^2$$

The table below shows the values of a, b, and c for each of the inverse trig functions and the equivalent expressions for $$\theta$$ that result from plugging the values into the equations $$\theta = -i\ln\left(\tfrac{a + ib}{c}\right)$$ above and simplifying.
 * $$\begin{align}

& a & & b & & c && -i\ln\left(\frac{a+ib}{c}\right) && \theta && \theta_{a,b\in\R}\\ \arcsin(z)\ \ & \sqrt{1 - z^2} & & z & & 1 & & -i\ln\left( \frac{\sqrt{1 - z^2} + iz}{1} \right) && = -i\ln\left( \sqrt{1 - z^2} + iz \right) && \operatorname{Im}\left(\ln\left( \sqrt{1 - z^2} + iz \right)\right) \\ \arccos(z)\ \ & z & & \sqrt{1 - z^2} & & 1 & & -i\ln\left( \frac{z + i\sqrt{1 - z^2}}{1} \right) && = -i\ln\left( z + \sqrt{z^2 - 1} \right) && \operatorname{Im}\left(\ln\left( z + \sqrt{z^2 - 1} \right)\right) \\ \arctan(z)\ \ & 1 & & z & & \sqrt{1 + z^2} & & -i\ln\left( \frac{1 + iz}{\sqrt{1 + z^2}} \right) && = -\frac{i}{2}\ln\left( \frac{i-z}{i+z} \right) && \operatorname{Im}\left(\ln\left( 1 + iz \right)\right) \\ \arccot(z)\ \ & z & & 1 & & \sqrt{z^2 + 1} & & -i\ln\left( \frac{z + i}{\sqrt{z^2 + 1}} \right) && = -\frac{i}{2}\ln\left( \frac{z + i}{z-i} \right) && \operatorname{Im}\left(\ln\left( z + i \right)\right) \\ \arcsec(z)\ \ & 1 & & \sqrt{z^2 - 1} & & z & & -i\ln\left( \frac{1 + i\sqrt{z^2 - 1}}{z} \right) && = -i\ln\left( \frac{1}{z} + \sqrt{\frac{1}{z^2}-1} \right) && \operatorname{Im}\left(\ln\left( \frac{1}{z} + \sqrt{\frac{1}{z^2}-1} \right)\right) \\ \arccsc(z)\ \ & \sqrt{z^2 - 1} & & 1 & & z & & -i\ln\left( \frac{\sqrt{z^2 - 1} + i}{z} \right) && = -i\ln\left( \sqrt{1-\frac{1}{z^2}} + \frac{i}{z} \right) && \operatorname{Im}\left(\ln\left( \sqrt{1 - \frac{1}{z^2}} + \frac{i}{z} \right)\right) \\ \end{align}$$

The particular form of the simplified expression can cause the output to differ from the usual principal branch of each of the inverse trig functions. The formulations given will output the usual principal branch when using the $$ \operatorname{Im}\left( \ln z \right) \in (-\pi,\pi] $$ and $$ \operatorname{Re}\left(\sqrt{z}\right) \ge 0 $$ principal branch for every function except arccotangent in the $$\theta$$ column. Arccotangent in the $$\theta$$ column will output on its usual principal branch by using the $$ \operatorname{Im}\left( \ln z \right) \in [0,2\pi) $$ and $$ \operatorname{Im}\left(\sqrt{z}\right) \ge 0 $$ convention.

In this sense, all of the inverse trig functions can be thought of as specific cases of the complex-valued log function. Since these definition work for any complex-valued $$z$$, the definitions allow for hyperbolic angles as outputs and can be used to further define the inverse hyperbolic functions. It's possible to algebraically prove these relations by starting with the exponential forms of the trigonometric functions and solving for the inverse function.

Example proof

 * $$\begin{align}

\sin(\phi) &= z \\ \phi &= \arcsin(z) \end{align}$$

Using the exponential definition of sine, and letting $$\xi = e^{i \phi}, $$

 * $$\begin{align}

z &= \frac{e^{i \phi} - e^{-i \phi}}{2i} \\[10mu] 2iz &= \xi - \frac{1}{\xi} \\[5mu] 0 &= \xi^2 - 2i z \xi - 1 \\[5mu] \xi &= iz \pm \sqrt{1 - z^2} \\[5mu] \phi &= -i \ln \left(iz \pm \sqrt{1 - z^2}\right) \end{align}$$

(the positive branch is chosen)

 * $$\phi= \arcsin(z) = -i \ln \left(iz + \sqrt{1-z^2} \right)$$

Finding the angle of a right triangle
Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine and cosine, it follows that

 * $$\theta = \arcsin \left( \frac{\text{opposite}}{\text{hypotenuse}} \right) = \arccos \left( \frac{\text{adjacent}}{\text{hypotenuse}} \right) .$$

Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the Pythagorean Theorem: $$a^2+b^2=h^2$$ where $$h$$ is the length of the hypotenuse. Arctangent comes in handy in this situation, as the length of the hypotenuse is not needed.

 * $$\theta = \arctan \left( \frac{\text{opposite}}{\text{adjacent}} \right) \, .$$

For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle θ with the horizontal, where θ may be computed as follows:

 * $$\theta

= \arctan \left( \frac{\text{opposite}}{\text{adjacent}} \right) = \arctan \left( \frac{\text{rise}}{\text{run}} \right) = \arctan \left( \frac{8}{20} \right) \approx 21.8^{\circ} \, .$$

Two-argument variant of arctangent
The two-argument atan2 function computes the arctangent of y / x given y and x, but with a range of (−$\pi$, π]. In other words, atan2(y, x) is the angle between the positive x-axis of a plane and the point (x, y) on it, with positive sign for counter-clockwise angles (upper half-plane, y > 0), and negative sign for clockwise angles (lower half-plane, y < 0). It was first introduced in many computer programming languages, but it is now also common in other fields of science and engineering. In terms of the standard arctan function, that is with range of (−$x$, $x$), it can be expressed as follows:

 * $$\operatorname{atan2}(y, x) = \begin{cases}

\arctan\left(\frac y x\right)      & \quad x > 0 \\ \arctan\left(\frac y x\right) + \pi & \quad y \ge 0,\; x < 0 \\ \arctan\left(\frac y x\right) - \pi & \quad y < 0,\; x < 0 \\ \frac{\pi}{2}  & \quad y > 0,\; x = 0 \\ -\frac{\pi}{2}  & \quad y < 0,\; x = 0 \\ \text{undefined} & \quad y = 0,\; x = 0 \end{cases}$$

It also equals the principal value of the argument of the complex number x + iy.

This limited version of the function above may also be defined using the tangent half-angle formulae as follows:
 * $$\operatorname{atan2}(y, x) = 2\arctan\left(\frac{y}{\sqrt{x^2 + y^2} + x}\right)$$

provided that either x > 0 or y ≠ 0. However this fails if given x ≤ 0 and y = 0 so the expression is unsuitable for computational use.

The above argument order (y, x) seems to be the most common, and in particular is used in ISO standards such as the C programming language, but a few authors may use the opposite convention (x, y) so some caution is warranted. These variations are detailed at atan2.

Arctangent function with location parameter
In many applications the solution $$y$$ of the equation $$x=\tan(y)$$ is to come as close as possible to a given value $$-\infty < \eta < \infty$$. The adequate solution is produced by the parameter modified arctangent function

y = \arctan_\eta(x) := \arctan(x) + \pi \, \operatorname{rni}\left(\frac{\eta - \arctan(x)}{\pi} \right)\,. $$

The function $$\operatorname{rni}$$ rounds to the nearest integer.

Numerical accuracy
For angles near 0 and π, arccosine is ill-conditioned, and similarly with arcsine for angles near −π/2 and π/2. Computer applications thus need to consider the stability of inputs to these functions and the sensitivity of their calculations, or use alternate methods.

title: Inverse trigonometric functions


text:
Triangular matrix

In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called  if all the entries above the main diagonal are zero. Similarly, a square matrix is called  if all the entries below the main diagonal are zero.

Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix L and an upper triangular matrix U if and only if all its leading principal minors are non-zero.

Description
A matrix of the form
 * $$L = \begin{bmatrix}

\ell_{1,1} &           &        &              &          0 \\ \ell_{2,1} & \ell_{2,2} &       &              &            \\ \ell_{3,1} & \ell_{3,2} & \ddots &             &            \\ \vdots &    \vdots & \ddots &       \ddots &            \\ \ell_{n,1} & \ell_{n,2} & \ldots & \ell_{n,n-1} & \ell_{n,n} \end{bmatrix}$$

is called a lower triangular matrix or left triangular matrix, and analogously a matrix of the form
 * $$U = \begin{bmatrix}

u_{1,1} & u_{1,2} & u_{1,3} & \ldots &  u_{1,n} \\ & u_{2,2} & u_{2,3} & \ldots &  u_{2,n} \\ &        &  \ddots & \ddots &    \vdots \\ &        &         & \ddots & u_{n-1,n} \\ 0 &        &         &        &   u_{n,n} \end{bmatrix}$$

is called an upper triangular matrix or right triangular matrix. A lower or left triangular matrix is commonly denoted with the variable L, and an upper or right triangular matrix is commonly denoted with the variable U or R.

A matrix that is both upper and lower triangular is diagonal. Matrices that are similar to triangular matrices are called triangularisable.

A non-square (or sometimes any) matrix with zeros above (below) the diagonal is called a lower (upper) trapezoidal matrix. The non-zero entries form the shape of a trapezoid.

Examples
The matrix

 * $$\begin{bmatrix}

1 & 0 & 0 \\ 2 & 96 & 0 \\  4 & 9 & 69 \\ \end{bmatrix}$$

is lower triangular, and

 * $$\begin{bmatrix}

1 & 4 & 1 \\ 0 & 6 & 9 \\  0 & 0 & 1 \\ \end{bmatrix}$$

is upper triangular.

Forward and back substitution
A matrix equation in the form $$L\mathbf{x} = \mathbf{b}$$ or $$U\mathbf{x} = \mathbf{b}$$ is very easy to solve by an iterative process called forward substitution for lower triangular matrices and analogously back substitution for upper triangular matrices. The process is so called because for lower triangular matrices, one first computes $$x_1$$, then substitutes that forward into the next equation to solve for $$x_2$$, and repeats through to $$x_n$$. In an upper triangular matrix, one works backwards, first computing $$x_n$$, then substituting that back into the previous equation to solve for $$x_{n-1}$$, and repeating through $$x_1$$.

Notice that this does not require inverting the matrix.

Forward substitution
The matrix equation Lx = b can be written as a system of linear equations

 * $$\begin{matrix}

\ell_{1,1} x_1 &  &                &   &        &   &                & = &    b_1 \\ \ell_{2,1} x_1 & + & \ell_{2,2} x_2 &  &        &   &                & = &    b_2 \\ \vdots &  &         \vdots &   & \ddots &   &                &   & \vdots \\ \ell_{m,1} x_1 & + & \ell_{m,2} x_2 & + & \dotsb & + & \ell_{m,m} x_m & = &   b_m \\ \end{matrix}$$

Observe that the first equation ($$\ell_{1,1} x_1 = b_1$$) only involves $$x_1$$, and thus one can solve for $$x_1$$ directly. The second equation only involves $$x_1$$ and $$x_2$$, and thus can be solved once one substitutes in the already solved value for $$x_1$$. Continuing in this way, the $$k$$-th equation only involves $$x_1,\dots,x_k$$, and one can solve for $$x_k$$ using the previously solved values for $$x_1,\dots,x_{k-1}$$. The resulting formulas are:
 * $$\begin{align}

x_1 &= \frac{b_1}{\ell_{1,1}}, \\ x_2 &= \frac{b_2 - \ell_{2,1} x_1}{\ell_{2,2}}, \\ &\ \ \vdots \\ x_m &= \frac{b_m - \sum_{i=1}^{m-1} \ell_{m,i}x_i}{\ell_{m,m}}. \end{align}$$

A matrix equation with an upper triangular matrix U can be solved in an analogous way, only working backwards.

Applications
Forward substitution is used in financial bootstrapping to construct a yield curve.

Properties
The transpose of an upper triangular matrix is a lower triangular matrix and vice versa.

A matrix which is both symmetric and triangular is diagonal. In a similar vein, a matrix which is both normal (meaning A*A = AA*, where A* is the conjugate transpose) and triangular is also diagonal. This can be seen by looking at the diagonal entries of A*A and AA*.

The determinant and permanent of a triangular matrix equal the product of the diagonal entries, as can be checked by direct computation.

In fact more is true: the eigenvalues of a triangular matrix are exactly its diagonal entries. Moreover, each eigenvalue occurs exactly k times on the diagonal, where k is its algebraic multiplicity, that is, its multiplicity as a root of the characteristic polynomial $$p_A(x)=\det(xI-A)$$ of A. In other words, the characteristic polynomial of a triangular n×n matrix A is exactly
 * $$p_A(x) = (x-a_{11})(x-a_{22})\cdots(x-a_{nn})$$,

that is, the unique degree n polynomial whose roots are the diagonal entries of A (with multiplicities). To see this, observe that $$xI-A$$ is also triangular and hence its determinant $$\det(xI-A)$$ is the product of its diagonal entries $$(x-a_{11})(x-a_{22})\cdots(x-a_{nn})$$.

Unitriangular matrix
If the entries on the main diagonal of a (upper or lower) triangular matrix are all 1, the matrix is called (upper or lower) unitriangular.

Other names used for these matrices are unit (upper or lower) triangular, or very rarely normed (upper or lower) triangular. However, a unit triangular matrix is not the same as the unit matrix, and a normed triangular matrix has nothing to do with the notion of matrix norm.

All finite unitriangular matrices are unipotent.

Strictly triangular matrix
If all of the entries on the main diagonal of a (upper or lower) triangular matrix are also 0, the matrix is called strictly (upper or lower) triangular.

All finite strictly triangular matrices are nilpotent of index at most n as a consequence of the Cayley-Hamilton theorem.

Atomic triangular matrix
An atomic (upper or lower) triangular matrix is a special form of unitriangular matrix, where all of the off-diagonal elements are zero, except for the entries in a single column. Such a matrix is also called a Frobenius matrix, a Gauss matrix, or a Gauss transformation matrix.

Block triangular matrix
A block triangular matrix is a block matrix (partitioned matrix) that is a triangular matrix.

Upper block triangular
A matrix $$A$$ is upper block triangular if

 * $$A = \begin{bmatrix}

A_{11} & A_{12} & \cdots & A_{1k} \\ 0 & A_{22} & \cdots & A_{2k} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & A_{kk} \end{bmatrix}$$,

where $$A_{ij} \in \mathbb{F}^{n_i \times n_j}$$ for all $$i, j = 1, \ldots, k$$.

Lower block triangular
A matrix $$A$$ is lower block triangular if

 * $$A = \begin{bmatrix}

A_{11} & 0 & \cdots & 0 \\ A_{21} & A_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ A_{k1} & A_{k2} & \cdots & A_{kk} \end{bmatrix}$$,

where $$A_{ij} \in \mathbb{F}^{n_i \times n_j}$$ for all $$i, j = 1, \ldots, k$$.

Triangularisability
A matrix that is similar to a triangular matrix is referred to as triangularizable. Abstractly, this is equivalent to stabilizing a flag: upper triangular matrices are precisely those that preserve the standard flag, which is given by the standard ordered basis $$(e_1,\ldots,e_n)$$ and the resulting flag $$0 < \left\langle e_1\right\rangle < \left\langle e_1,e_2\right\rangle < \cdots < \left\langle e_1,\ldots,e_n \right\rangle = K^n.$$ All flags are conjugate (as the general linear group acts transitively on bases), so any matrix that stabilises a flag is similar to one that stabilizes the standard flag.

Any complex square matrix is triangularizable. In fact, a matrix A over a field containing all of the eigenvalues of A (for example, any matrix over an algebraically closed field) is similar to a triangular matrix. This can be proven by using induction on the fact that A has an eigenvector, by taking the quotient space by the eigenvector and inducting to show that A stabilizes a flag, and is thus triangularizable with respect to a basis for that flag.

A more precise statement is given by the Jordan normal form theorem, which states that in this situation, A is similar to an upper triangular matrix of a very particular form. The simpler triangularization result is often sufficient however, and in any case used in proving the Jordan normal form theorem.

In the case of complex matrices, it is possible to say more about triangularization, namely, that any square matrix A has a Schur decomposition. This means that A is unitarily equivalent (i.e. similar, using a unitary matrix as change of basis) to an upper triangular matrix; this follows by taking an Hermitian basis for the flag.

Simultaneous triangularisability
A set of matrices $$A_1, \ldots, A_k$$ are said to be  if there is a basis under which they are all upper triangular; equivalently, if they are upper triangularizable by a single similarity matrix P. Such a set of matrices is more easily understood by considering the algebra of matrices it generates, namely all polynomials in the $$A_i,$$ denoted $$K[A_1,\ldots,A_k].$$ Simultaneous triangularizability means that this algebra is conjugate into the Lie subalgebra of upper triangular matrices, and is equivalent to this algebra being a Lie subalgebra of a Borel subalgebra.

The basic result is that (over an algebraically closed field), the commuting matrices $$A,B$$ or more generally $$A_1,\ldots,A_k$$ are simultaneously triangularizable. This can be proven by first showing that commuting matrices have a common eigenvector, and then inducting on dimension as before. This was proven by Frobenius, starting in 1878 for a commuting pair, as discussed at commuting matrices. As for a single matrix, over the complex numbers these can be triangularized by unitary matrices.

The fact that commuting matrices have a common eigenvector can be interpreted as a result of Hilbert's Nullstellensatz: commuting matrices form a commutative algebra $$K[A_1,\ldots,A_k]$$ over $$K[x_1,\ldots,x_k]$$ which can be interpreted as a variety in k-dimensional affine space, and the existence of a (common) eigenvalue (and hence a common eigenvector) corresponds to this variety having a point (being non-empty), which is the content of the (weak) Nullstellensatz. In algebraic terms, these operators correspond to an algebra representation of the polynomial algebra in k variables.

This is generalized by Lie's theorem, which shows that any representation of a solvable Lie algebra is simultaneously upper triangularizable, the case of commuting matrices being the abelian Lie algebra case, abelian being a fortiori solvable.

More generally and precisely, a set of matrices $$A_1,\ldots,A_k$$ is simultaneously triangularisable if and only if the matrix $$p(A_1,\ldots,A_k)[A_i,A_j]$$ is nilpotent for all polynomials p in k non-commuting variables, where $$[A_i,A_j]$$ is the commutator; for commuting $$A_i$$ the commutator vanishes so this holds. This was proven by Drazin, Dungey, and Gruenberg in 1951; a brief proof is given by Prasolov in 1994. One direction is clear: if the matrices are simultaneously triangularisable, then $$[A_i, A_j]$$ is strictly upper triangularizable (hence nilpotent), which is preserved by multiplication by any $$A_k$$ or combination thereof – it will still have 0s on the diagonal in the triangularizing basis.

Algebras of triangular matrices

Upper triangularity is preserved by many operations:
 * The sum of two upper triangular matrices is upper triangular.
 * The product of two upper triangular matrices is upper triangular.
 * The inverse of an upper triangular matrix, if it exists, is upper triangular.
 * The product of an upper triangular matrix and a scalar is upper triangular.

Together these facts mean that the upper triangular matrices form a subalgebra of the associative algebra of square matrices for a given size. Additionally, this also shows that the upper triangular matrices can be viewed as a Lie subalgebra of the Lie algebra of square matrices of a fixed size, where the Lie bracket [a, b] given by the commutator ab − ba. The Lie algebra of all upper triangular matrices is a solvable Lie algebra. It is often referred to as a Borel subalgebra of the Lie algebra of all square matrices.

All these results hold if upper triangular is replaced by lower triangular throughout; in particular the lower triangular matrices also form a Lie algebra. However, operations mixing upper and lower triangular matrices do not in general produce triangular matrices. For instance, the sum of an upper and a lower triangular matrix can be any matrix; the product of a lower triangular with an upper triangular matrix is not necessarily triangular either.

The set of unitriangular matrices forms a Lie group.

The set of strictly upper (or lower) triangular matrices forms a nilpotent Lie algebra, denoted $$\mathfrak{n}.$$ This algebra is the derived Lie algebra of $$\mathfrak{b}$$, the Lie algebra of all upper triangular matrices; in symbols, $$\mathfrak{n} = [\mathfrak{b},\mathfrak{b}].$$ In addition, $$\mathfrak{n}$$ is the Lie algebra of the Lie group of unitriangular matrices.

In fact, by Engel's theorem, any finite-dimensional nilpotent Lie algebra is conjugate to a subalgebra of the strictly upper triangular matrices, that is to say, a finite-dimensional nilpotent Lie algebra is simultaneously strictly upper triangularizable.

Algebras of upper triangular matrices have a natural generalization in functional analysis which yields nest algebras on Hilbert spaces.

Borel subgroups and Borel subalgebras
The set of invertible triangular matrices of a given kind (upper or lower) forms a group, indeed a Lie group, which is a subgroup of the general linear group of all invertible matrices. A triangular matrix is invertible precisely when its diagonal entries are invertible (non-zero).

Over the real numbers, this group is disconnected, having $$2^n$$ components accordingly as each diagonal entry is positive or negative. The identity component is invertible triangular matrices with positive entries on the diagonal, and the group of all invertible triangular matrices is a semidirect product of this group and the group of diagonal matrices with $$\pm 1$$ on the diagonal, corresponding to the components.

The Lie algebra of the Lie group of invertible upper triangular matrices is the set of all upper triangular matrices, not necessarily invertible, and is a solvable Lie algebra. These are, respectively, the standard Borel subgroup B of the Lie group GLn and the standard Borel subalgebra $$\mathfrak{b}$$ of the Lie algebra gln.

The upper triangular matrices are precisely those that stabilize the standard flag. The invertible ones among them form a subgroup of the general linear group, whose conjugate subgroups are those defined as the stabilizer of some (other) complete flag. These subgroups are Borel subgroups. The group of invertible lower triangular matrices is such a subgroup, since it is the stabilizer of the standard flag associated to the standard basis in reverse order.

The stabilizer of a partial flag obtained by forgetting some parts of the standard flag can be described as a set of block upper triangular matrices (but its elements are not all triangular matrices). The conjugates of such a group are the subgroups defined as the stabilizer of some partial flag. These subgroups are called parabolic subgroups.

Examples
The group of 2×2 upper unitriangular matrices is isomorphic to the additive group of the field of scalars; in the case of complex numbers it corresponds to a group formed of parabolic Möbius transformations; the 3×3 upper unitriangular matrices form the Heisenberg group.

title: Triangular matrix


text:
Slender-billed curlew

The slender-billed curlew (Numenius tenuirostris) is a bird in the wader family Scolopacidae. Isotope analysis suggests the majority of the former population bred in the Kazakh Steppe despite a record from the Siberian swamps, and was migratory, formerly wintering in shallow freshwater habitats around the Mediterranean. This species has occurred as a vagrant in western Europe, the Canary Islands, the Azores, Oman, Canada, and Japan. The slender-billed curlew was always a rare species and is feared extinct, with the last verifiable sighting being in 2004.

Description
The slender-billed curlew is a small curlew, 36 – in length with a 77 – wingspan. It is therefore about the same size as a Eurasian whimbrel, but it is more like the Eurasian curlew in plumage. The breeding adult is mainly greyish brown above, with a whitish rump and lower back. The underparts are whitish, heavily streaked with dark brown. The flanks have round or heart-shaped spots. The non-breeding plumage is similar, but with fewer flank spots. Male and female are alike in plumage, but females are longer-billed than males, an adaptation in curlew species that eliminates direct competition for food between the sexes. The juvenile plumage is very similar to the adult, but the flank are marked with brown streaking, the heart-shaped spots only appearing toward the end of the first winter.

Compared to the Eurasian curlew, the slender-billed curlew is whiter on the breast, tail, and underwing, and the bill is shorter, more slender, and slightly straighter at the base. The arrowhead-shaped flank spots of the Eurasian curlew also are different from the round or heart-shaped spots of the adults slender-billed. The head pattern, with a dark cap and whitish supercilium, recalls that of the whimbrel, but that species also has a central crown stripe and a more clearly marked pattern overall; the pattern of the slender-billed curlew would be hard to make out in the field.

This species shows more white than other curlews; however, the white underwing has been stressed too much as a relevant identification criteria along with the distinctive flank markings of adults (not helpful in juvenile and 1st year bird before post-juvenile moult). The most recent and most updated identification's paper, reports as clinching characters the uniformly dark underside of 4 to 6 outer primaries (the wing-tip or "hand"), the black (adults) or anyway daker (juvenile and 1st y bird) legs, and the white tail with fewer dark bars (Corso et al., 2014).

Vocalisations
The call is a cour-lee, similar to that of the Eurasian curlew, but higher-pitched, more melodic, and shorter. The alarm call is a fast cu-ee.

Distribution
The slender-billed curlew was only known to breed in a small region north of Omsk, Russia in a period between 1909 and 1925. In recent history, it mostly migrated to the Mediterranean as well as southern Arabia, with claims in the northern reaches of the Persian gulf, in Kuwait and Iraq. There have been historical records of the bird elsewhere, as in an ornithological dictionary of Gibraltar, written in 1895, it indirectly states that the slender-billed curlew was recorded as a passage migrant in Malaga, Spain.

Behaviour
Little is known about the breeding biology, but on average the few nests observed had four eggs.

Slender-billed curlews feed by using their bills to probe soft mud for small invertebrates, but will also pick other small items off the surface if the opportunity arises. It used to be highly gregarious outside the breeding season, associating with related species, particularly Eurasian curlews.

Status
After a long period of steady decline, the slender-billed curlew is extremely rare, with only a minute and still declining population. This is thought to be fewer than 50 adult birds, with the last verified sighting in 2004. As a result, it is now listed as critically endangered. However, for most specialists of the species, it is most probably extinct now (Corso et al., 2014; Kirwan et al., 2015). Indeed, all records, away from the famous ones from Morocco, from 1990 until today are not considered reliable and acceptable on stricter criteria.

The primary cause of the decline is thought to be excessive hunting on the Mediterranean wintering grounds. Habitat loss, particularly in the wintering grounds, may also have played a part, but huge areas of forest bogs suitable for breeding still exist in Siberia. It is unknown to what extent the birds still reproduce successfully, and how much gene flow still exists in what may once have been a large and widely dispersed population undergoing little purging of deleterious recessive alleles and consequently with a high MVP. Furthermore, there is evidence that birds in winter quarters were more numerous once, and in general not a very rare sight in Western Europe in the nineteenth century, where they were hunted with some regularity. Later on they were additionally threatened by pollution, e.g. oil spills. There are no data about how these threats endanger the species today. Theoretically, they might have retreated to all but inaccessible areas, but then, a single hunter or fox might unwittingly wipe out enough of the few remaining birds to doom the species.

The last well-documented nest was found in 1924, near Tara in Omsk oblast, Siberia (57°N, 74°W). Its nesting grounds since then remain unknown, despite several intensive searches (not surprising, with more than 100,000 square kilometres to search). The extent of its decline also is reflected in the absence of wintering birds at previously regular Moroccan sites.

More recently, 20 birds were recorded in Italy in 1995, but this most unbelievable record is now confirmed to be referred to Numenius arquata orientalis as both photographs and sound recordings shown (Kirwan et al., 2015). There was a potential record of an immature (one year old) at Druridge Pools in Northumberland, England, on 4–7 May 1998, for details of which see the Druridge Bay curlew. The bird was initially accepted onto the British List but was removed in 2013 following a review of the identification.

Slender-billed curlews have been reported in various Western Palearctic locations on a number of occasions since the Druridge bird, including claimed, but unverified, sightings of single birds from Italy and Greece; none have been documented with conclusive photographs and at least one claimed bird, at RSPB Minsmere, Suffolk, England, in 2004, is now widely believed to have been a Eurasian curlew. Further sourced reports of the species were published in 2007, in British Birds magazine; the article stated, quoting from Zhmud:

"During the last few years, small groups of birds have been found in the northern coastal areas [of the Danube Delta], frequenting low-lying islands, bays, and sand-spits covered with Common Glasswort Salicornia europaea [...] Four birds were present from 25 July to 21 August 2003, six were seen on 11 August 2004, and another on 12 August 2004."

A sighting of a single bird was reported from Albania in 2006 by a team including ornithologists from the environmental organization EuroNatur.

Thus, although hard proof is lacking, but given the extent of possible habitat and the precautionary principle, it is believed to be extant for the time being. Apparently at least the wintering range has starkly contracted; it appears that the handful of family or neighbour groups that are left retreat to remote habitat in southeastern Europe in winter. The IUCN classifies it as Critically Endangered (CR) C2a(ii); D. This means that an estimated 50 mature birds or fewer are believed to exist, with numbers declining, and that there probably is only one subpopulation.

General references

 * Hayman, Peter; Marchant, John & Prater, Tony (1986). Shorebirds: an identification guide to the waders of the world. Houghton Mifflin, Boston. ISBN 0-395-60237-8

 * Svensson, Lars; Zetterström, Dan; Mullarney, Killian & Grant, P. J. (1999). Collins bird guide. Harper & Collins, London. ISBN 0-00-219728-6
 * Corso, Andrea; Jansen, Justin; Kokay, Szabolcs (2014). "A review of the identification criteria and variability of the Slender-billed Curlew." British Bird 107: 339–370.  https://www.researchgate.net/publication/269160426_A_review_of_the_identification_criteria_and_variability_of_the_Slender-billed_Curlew

Identification
Corso, Andrea; Jansen, Justin; Kokay, Szabolcs (2014). "A review of the identification criteria and variability of the Slender-billed Curlew." British Bird 107: 339–370. https://www.researchgate.net/publication/269160426_A_review_of_the_identification_criteria_and_variability_of_the_Slender-billed_Curlew
 * Identification of Slender-billed Curlew, John Marchant, British Birds 77: 135–140.
 * Slender-billed Curlew studies, Richard Porter, British Birds 77: 581–586.
 * Habitat of Slender-billed Curlews in Morocco, Arnoud van den Berg, British Birds 83: 1–7.
 * Slender-billed Curlew in Tunisia in Feb 1984, Eddy Wijmengs & Klaas van Dijk, Dutch Birding 7: 67–68.
 * Slender-billed Curlews in Morocco in Feb 1979, Peter Ewins, Dutch Birding 11: 119–120.
 * Identification of Slender-billed Curlew and its occurrence in Morocco in winter 1987/88, Arnoud van den Berg, Dutch Birding 10: 45–53.
 * Slender-billed Curlew on Sicily in March 1996, Andrea Corso, Dutch Birding 18: 302.
 * Slender-billed Curlew collected at Canis-vliet in September 1896, Gunter De Smet, Dutch Birding 19: 230–232.
 * The identification of the Slender-billed Curlew, British Birds Vol 56 No8 1963
 * Kirwan, Guy; Porter, Richard; Scott, Derek (2015). Chronicle of an extinction? A review of Slender-billed Curlew records in the Middle East. "British Birds" 108: 669–682.

title: Slender-billed curlew


text:
J. D. Tippit

J. D. Tippit (September 18, 1924 – November 22, 1963) was an American World War II U.S. Army veteran and Bronze Star recipient, who was a police officer with the Dallas Police Department for 11 years. On November 22, 1963, less than one hour after the assassination of President John F. Kennedy, Tippit was shot to death in a residential neighborhood in the Oak Cliff section of Dallas, Texas. Lee Harvey Oswald was initially arrested for the murder of Tippit and subsequently charged for killing Kennedy but was murdered by Jack Ruby before he could stand trial.

In 1964, the Warren Commission concluded that Tippit was murdered by Oswald and that Oswald was the sole perpetrator. The commission's findings remain controversial and has been both challenged by conspiracy theorists who dispute the commission's conclusions and supported by others.

Early life
J. D. Tippit was born near the town of Annona, Texas, in Red River County. He was the eldest of seven children to Edgar Lee Tippit (1902–2006) and Lizzie Mae "May Bug" Rush (1905–1990). The Tippit and Burns families were of English ancestry, their ancestors having immigrated to Virginia from England by 1635. It is sometimes reported that J. D. stood for "Jefferson Davis", but the letters did not stand for anything.

Tippit attended public schools through the tenth grade and was raised as a Baptist, a faith he practiced for the rest of his life. In the fall of 1939, when he was 15 years old, his family moved to Baker Lane, a stretch of dirt road six miles southwest from Clarksville, Texas.

Military service
He served in World War II, entering the United States Army on July 21, 1944. He volunteered for the paratroopers, part of the newly-formed airborne forces. After finishing his training, he was sent to Europe, in January 1945, and was assigned to the 513th Parachute Infantry Regiment (513th PIR), part of the 17th Airborne Division, which had recently fought in the Battle of the Bulge and suffered heavy casualties. He saw combat in Operation Varsity, the airborne crossing of the Rhine River in March 1945, earning a Bronze Star, and remained on active duty until June 20, 1946.

Career
Tippit began working for the Dearborn Stove Company in 1946. He next worked for Sears, Roebuck and Company in the installation department from March 1948 to September 1949 when he was laid off. Tippit and his wife Marie moved to Lone Star, Texas, where Tippit attempted to farm and raise cattle.

In January 1950, Tippit enrolled in a Veterans Administration vocational training school at Bogata, Texas. He left the school in June 1952. After several setbacks as a farmer and rancher, Tippit decided to become a police officer. The Tippit family then relocated to Dallas where Tippit was hired by the Dallas Police Department as a patrolman in July 1952. During his time with the Dallas Police Department, Tippit was cited twice for bravery.

At the time of his death, Tippit was earning a monthly salary of $490,, as a Dallas police officer. He was working two part-time jobs. He worked at Austin's Barbecue restaurant on Friday and Saturday nights and at the Stevens Park Theatre on Sundays.

Murder and investigation
On November 22, 1963, Tippit was working beat number 78, his normal patrol area in south Oak Cliff, a residential area of Dallas.

At 12:45 p.m., 15 minutes after President Kennedy was shot, Tippit received a radio order to drive to the central Oak Cliff area as part of a concentration of police around the center of the city. At 12:54, Tippit radioed that he had moved as directed. By then, several messages had been broadcast describing a suspect in the President's shooting at Dealey Plaza as a slender white male, in his early 30s, 5 ft tall, and weighing about 165 lb. Oswald was a slender white male, 24 years old, 5 ft tall, and an estimated weight of 135 lb at autopsy. Tippit spoke his last known words, "10-4", over his police radio.

According to the Warren Commission, at approximately 1:11–1:14 p.m., Tippit was driving slowly eastward on East 10th Street — about 100 ft past the intersection of 10th Street and Patton Avenue — when he pulled alongside a man who resembled the police description. Oswald walked over to Tippit's car and exchanged words with him through an open vent window. Tippit opened his car door and as he (Tippit) walked toward the front of the car, Oswald drew his handgun and fired four shots in rapid succession. Three bullets hit Tippit in the chest and as he lay to the ground, another shot hit his right temple, killing him instantly. Tippit's body was transported from the scene of the shooting by ambulance to Methodist Hospital, where he was pronounced dead at 1:25 p.m. by Dr. Richard A. Liguori.

A short time later, Hardy's shoe store manager Johnny Brewer observed Oswald acting suspiciously as police cars passed nearby with sirens blaring. Oswald then ducked into the Texas Theatre without purchasing a ticket. The police were notified by the theater's cashier and responded by surrounding the theater. Oswald was arrested after a brief struggle. Twelve people who witnessed the shooting or its aftermath were mentioned in the Warren Report.

Domingo Benavides saw Tippit standing by the left door of his parked police car, and a man standing on the right side of the car. He then heard three shots and saw Tippit fall to the ground. Benavides stopped his pickup truck on the opposite side of the street from Tippit's car. He observed the shooter fleeing the scene and removing two spent cartridge cases from his gun as he left.

Benavides waited in his truck until the gunman disappeared, and then "a few minutes" more, before assisting Tippit. He then tried, unsuccessfully, to use the radio in Tippit's car to report the shooting to police headquarters. Then another, unidentified person used the radio in the car and reported the shooting to a police operator for the first time. After that, Ted Callaway, who was Benavides' boss at the used car lot and a former Marine, used the radio and reported the shooting, hearing in response that the police already knew about it.

Callaway testified that he had seen the shooter with the gun "in a raised pistol position", and shouted at him, but what the shooter responded was unintelligible. Helen Markham witnessed the shooting and then saw a man with a gun in his hand leave the scene. Markham identified Oswald as Tippit's killer in a police lineup she viewed that evening.

Barbara Davis and her sister-in-law Virginia Davis heard the shots and saw a man crossing their lawn, shaking his revolver, as if he were emptying it of cartridge cases. Later, the women found two cartridge cases near the crime scene and handed the cases over to police. Two other cartridge cases were handed to a policeman by Benavides. That evening, Barbara Davis and Virginia Davis were taken to a lineup and both Davises picked out Oswald as the man whom they had seen.

Taxicab driver William Scoggins testified that he was sitting nearby in his cab when he saw Tippit's police car pull up alongside a man on the sidewalk. Scoggins heard three or four shots and then saw Tippit fall to the ground. As Scoggins crouched behind his cab, the man passed within 12 feet of him, pistol in hand, muttering what sounded to him like, "poor dumb cop" or "poor damn cop." The next day, Scoggins viewed a police lineup and identified Oswald as the man whom he had seen with the pistol.

The Commission also named several other witnesses who were not at the scene of the murder, but who identified Oswald running between the murder scene and the Texas Theatre, where Oswald was subsequently arrested.

It was the unanimous testimony of expert witnesses before the Warren Commission that these spent cartridge cases were fired from the revolver in Oswald's possession to the exclusion of all other weapons.

Out of the four bullets recovered from Tippit's body, only one (according to Nicol) or none (according to Cunningham) could be positively identified as having been fired from Oswald's revolver; the others "could have" been fired from that revolver, but there was no certain match. When the revolver was test-fired by the FBI, it was reported that it was leaving inconsistent microscopic markings on the bullets, i.e. two consecutive bullets fired from it could not be matched to each other. This was because the revolver had been rechambered for .38 Special but not rebarreled for .38 Special, so the bullets were slightly undersized compared to the barrel, making their passage through the barrel "erratic".

Extensive damage to the bullets, mutilation, was noted. Later, the House Select Committee on Assassinations (HSCA) agreed with Cunningham's conclusion that none of the bullets found could be positively identified, or ruled out, as having been fired from Oswald's revolver. Still, when they test-fired the gun, they found that bullets fired from it could actually be matched to each other, if they were of the same type and manufacturer.

There was a discrepancy between the four cartridge cases (2 Western, 2 Remington-Peters) and the four bullets (3 Western-Winchester, 1 Remington-Peters) found; one of the proposed explanations was that Oswald fired five shots, and one bullet and one cartridge case were not found.

Upon his arrest and during subsequent questionings by police, Oswald denied any involvement in Tippit's murder. Based on eyewitness' statements and the gun found in Oswald's possession at the time of his arrest, he was formally charged with the murder of Tippit at 7:10 p.m. on November 22. During the course of the day, police began to suspect that Oswald was also involved in the shooting of Kennedy. At approximately 1:00 am on November 23, Oswald was also charged with assassinating President John F. Kennedy. Oswald continued to maintain his innocence in connection with both murders. On November 24, while being transported from the Dallas City Jail to the Dallas County Jail, Oswald was shot and mortally wounded by Dallas nightclub owner Jack Ruby. The shooting was broadcast throughout the United States and Canada on live television.

As Oswald was killed before he was tried for either crime, President Lyndon B. Johnson commissioned a committee of US Senators, Congressmen and elder statesmen to investigate the events surrounding the deaths of Kennedy, Tippit, and Oswald in an effort to answer questions regarding the events. President Johnson also hoped to quell rumors that arose after Oswald was shot by Ruby that the assassination and subsequent shootings were part of a conspiracy. The committee, known as the Warren Commission, named for the commission chairman, Chief Justice Earl Warren, spent ten months investigating the murders and interviewing witnesses. On September 24, 1964, the Warren Commission released an 888-page report that concluded there was no evidence of a conspiracy and Oswald had acted alone in killing Kennedy and Tippit. The report also concluded that Ruby acted alone in the killing of Oswald.

In 1979, the HSCA reported: "Based on Oswald's possession of the murder weapon a short time after the murder and the eyewitness identifications of Oswald as the gunman, the committee concluded that Oswald shot and killed Tippit."

Conspiracy theories
Some conspiracy theorists have alleged that the murder of Tippit was part of a conspiracy to kill Kennedy, implying that two murders could not have happened so closely together by coincidence. Warren Commission attorney David Belin referred to Tippit's killing as the "Rosetta Stone to the JFK assassination". Other conspiracy theorists suggest that Tippit's murder was unconnected to the assassination.

Some conspiracy theorists dispute that Oswald shot Tippit, alleging that the physical evidence and witness testimony do not support that conclusion. New Orleans district attorney Jim Garrison, who investigated the assassination of John F. Kennedy and brought evidence in his 1969 trial of businessman Clay Shaw, contended in his book On the Trail of the Assassins that the witness testimony and handling of evidence in the Tippit murder was flawed and that it was doubtful that Oswald was the killer or even at the scene of the crime. According to Garrison, numerous witnesses who were not interviewed by the Warren Commission reported seeing two men fleeing the scene of Tippit's murder.

Garrison claimed that Helen Markham, the Warren Commission's star witness, expressed uncertainty as to her identification of Oswald in the police lineup. Garrison claimed that bullets recovered from Tippit's body were from two different manufacturers, as the Warren Commission stated, and the gun found on Oswald at his arrest did not match the cartridges found at the scene. Garrison accused the Dallas Police Department of mishandling the evidence and of possibly firing Oswald's revolver to produce bullet cartridges for the FBI to link to his gun.

Other conspiracy theorists allege that Tippit himself was a conspirator, tasked to kill Oswald by organized crime or right-wing politicians in order to cover up the search for other assassins.

Aftermath
On the evening of the assassination, both Attorney General Robert F. Kennedy and the new President, Lyndon B. Johnson, called Tippit's widow to express their sympathies. Jacqueline Kennedy wrote a letter expressing sorrow for the bond they shared. The plight of Tippit's family also moved much of the nation and a total of $647,579 was donated to them,, following the assassination. One of the largest individual gifts was $25,000 that Dallas businessman Abraham Zapruder donated to Marie Tippit after selling his film of the president's assassination to Life magazine.

A televised funeral service for Tippit was held on November 25, 1963, at the Beckley Hills Baptist Church, attended by about 2,000 people, at least 800 of them police colleagues. Police outriders attended the hearse on its way to the burial at the newly established Memorial Court of Honor at the Laurel Land Memorial Park in Dallas. His funeral was held on the same day as those of both Kennedy and Oswald.

In January 1964, Tippit was posthumously awarded the Medal of Valor from the American Police Hall of Fame, and he also received the Police Medal of Honor, the Police Cross, and the Citizens Traffic Commission Award of Heroism. A state historical marker memorializing Tippit was unveiled November 20, 2012, at the location where the shooting occurred.

Tippit's widow married Dallas police lieutenant Harry Dean Thomas in January 1967. They were married until his death in 1982. Marie Tippit later married Carl Flinner; the marriage ended in divorce after which Marie resumed using the surname of Tippit.

Personal life
Tippit married Marie Frances Gasway on December 26, 1946, and the couple had three children: Charles Allan (1950–2014), Brenda Kay (born 1953), and Curtis Glenn (born 1958). Marie died at 92 on March 2, 2021, after suffering from COVID-19 and other medical issues.

In popular culture
Tippit has been portrayed by Price Carson in 1991's JFK, David Duchovny in 1992's Ruby, and Matt Micou in 2013's Killing Kennedy. He was portrayed by uncredited actors in 1977's The Trial of Lee Harvey Oswald and 1978's Ruby and Oswald.

title: J. D. Tippit


text:
Count of Champagne

The Count of Champagne was the ruler of the County of Champagne from 950 to 1316. Champagne evolved from the County of Troyes in the late eleventh century and Hugh I was the first to officially use the title count of Champagne.

Count Theobald IV of Champagne inherited the Kingdom of Navarre in 1234. His great-granddaughter Joan married King Philip IV of France. Upon Joan's death in 1305, their son Louis became the last independent count of Champagne, with the title merging into the royal domain upon his accession to the French throne in 1314.

The titular counts of Champagne also inherited the post of seneschal of France.

Dukes of Champagne
In Merovingian and Carolingian times, several dukes of Champagne (or Campania) are known. The duchy appears to have been created by combining the civitates of Rheims, Châlons-sur-Marne, Laon, and Troyes. In the late seventh and early eighth centuries, Champagne was controlled by the Pippinids; first by Drogo, son of Pippin of Herstal, and then by Drogo's son Arnulf.

 * Drogo (690–707), abdicated
 * Arnulf (707-723), deposed

Counts of Meaux and Troyes
Counts of Troyes and Meaux
 * Robert of Troyes (956–967)
 * Herbert III of Meaux, (967–995)
 * Stephen I (995–1022)
 * Odo I of Meaux and III of Troyes (1022–1037), also Count of Blois
 * Stephen II (1037–1048)
 * Odo II of Meaux and IV of Troyes (1048–1066)
 * Theobald I (1066–1089), also Count of Blois

Counts of Champagne

 * Hugh (1102–1125)
 * Theobald II (1125–1152)
 * Henry I (1152–1181)
 * Henry II (1181–1197), also King of Jerusalem as Henry I
 * Theobald III (1197–1201)

 * Theobald IV, also King of Navarre as Theobald I (1201–1253)
 * Theobald V, also King of Navarre as Theobald II (1253–1270)
 * Henry III, also King of Navarre as Henry I (1270–1274)
 * Joan (1274–1305), Countess suo jure, also Queen of Navarre

 * Philip (co-ruler of Joan), also King of France as Philip IV
 * Louis (1305–1316), also King of Navarre, became King of France in 1314, after which the title merged into the royal domain

title: Count of Champagne


text:
MathWorld

MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana–Champaign.

History
Eric W. Weisstein, the creator of the site, was a physics and astronomy student who got into the habit of writing notes on his mathematical readings. In 1995 he put his notes online and called it "Eric's Treasure Trove of Mathematics." It contained hundreds of pages/articles, covering a wide range of mathematical topics. The site became popular as an extensive single resource on mathematics on the web. In 1998, he made a contract with CRC Press and the contents of the site were published in print and CD-ROM form, titled "CRC Concise Encyclopedia of Mathematics." The free online version became only partially accessible to the public. In 1999 Weisstein went to work for Wolfram Research, Inc. (WRI), and WRI renamed the Math Treasure Trove to MathWorld and hosted it on the company's website without access restrictions.

CRC lawsuit
In 2000, CRC Press sued Wolfram Research Inc. (WRI), WRI president Stephen Wolfram, and author Eric W. Weisstein, due to what they considered a breach of contract: that the MathWorld content was to remain in print only. The site was taken down by a court injunction.

The case was later settled out of court, with WRI paying an unspecified amount and complying with other stipulations. Among these stipulations is the inclusion of a copyright notice at the bottom of the website and broad rights for the CRC Press to produce MathWorld in printed book form. The site then became once again available free to the public.

This case made a wave of headlines in online publishing circles. The PlanetMath project was a result of MathWorld's being unavailable.

title: MathWorld