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| {{Trigonometry}}
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| In [[mathematics]], the '''inverse trigonometric functions''' (occasionally called '''cyclometric functions'''<ref>For example {{cite book |title=Triumph der Mathematik
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| |first=Heinrich|last=Dörrie|others=Trans. David Antin
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| |publisher=Dover|year=1965|isbn=0-486-61348-8|page=69}}</ref>) are the [[inverse function]]s of the [[trigonometric functions]] (with suitably restricted [[Domain of a function|domain]]s). Specifically, they are the inverses of the [[sine]], [[cosine]], [[tangent]], [[cotangent]], [[secant]], and [[cosecant]] functions. They 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]].
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| ==Notation==
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| There are many notations used for the inverse trigonometric functions. The notations {{math|sin<sup>−1</sup> (''x'')}}, {{math|cos<sup>−1</sup> (''x'')}}, {{math|tan<sup>−1</sup> (''x'')}}, etc. are often used, but this convention logically conflicts with the common semantics for expressions like {{math|sin<sup>2</sup> (''x'')}}, which refer to numeric power rather than function composition, and therefore may result in confusion between [[multiplicative inverse]] and [[Inverse function|compositional inverse]]. Another convention used by some authors<ref name=Book>{{cite book|title=Calculus and Analytic Geometry|year=1999|publisher=Punjab Textbook Board|location=[[Lahore]]|page=140|author=Prof. Sanaullah Bhatti|edition=First|coauthors=Ch. Nawab-ud-Din, Ch. Bashir Ahmed, Dr. S. M. Yousuf, Dr. Allah Bukhsh Taheem|editor=Prof. Mohammad Maqbool Ellahi, Dr. Karamat Hussain Dar, Faheem Hussain|language=[[Pakistani English]]|chapter=Differentiation of Tigonometric, Logarithmic and Exponential Functions}}</ref> is to use a [[majuscule]] (capital/upper-case) first letter along with a −1 superscript, e.g., {{math|Sin<sup>−1</sup> (''x'')}}, {{math|Cos<sup>−1</sup> (''x'')}}, etc., which avoids confusing them with the multiplicative inverse, which should be represented by {{math|sin<sup>−1</sup> (''x'')}}, {{math|cos<sup>−1</sup> (''x'')}}, etc. Yet another convention is to use an arc- prefix, so that the confusion with the −1 superscript is resolved completely, e.g., {{math|arcsin (''x'')}}, {{math|arccos (''x'')}}, etc. This convention is used throughout this article. In computer programming languages the inverse trigonometric functions are usually called asin, acos, atan.
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| ===Etymology of the arc- prefix===
| | Howdy. The author's name is Dalton but it's not the a lot of [http://Photobucket.com/images/masucline masucline] name out now there are. To drive is one of a person's things he loves mainly. His wife and him chose to exist in in South Carolina in addition , his family loves them. Auditing is where his primary income originates from. He has been running and maintaining one specific blog here: http://prometeu.net<br><br>Stop by my web site ... clash of clans hack android ([http://prometeu.net Read the Full Article]) |
| 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 arc whose cosine is x" is the same as "the angle whose cosine is x", because the measure of the length of the arc of the circle in radii is the same as the measurement of the angle in radians.
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| ==Principal values==
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| Since none of the six trigonometric functions are [[One-to-one function|one-to-one]], they are restricted in order to have inverse functions. Therefore the [[Range (mathematics)|range]]s of the inverse functions are proper [[subset]]s of the domains of the original functions
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| For example, using ''function'' in the sense of [[multivalued function]]s, just as the [[square root]] function ''y'' = √{{overline|''x''}} could be defined from ''y''<sup>2</sup> = ''x'', the function ''y'' = arcsin(''x'') is defined so that sin(''y'') = ''x''. There are multiple numbers ''y'' such that sin(''y'') = ''x''; for example, sin(0) = 0, but also sin({{pi}}) = 0, sin(2{{pi}}) = 0, etc. It follows that the arcsine function is [[Multivalued function|multivalued]]: arcsin(0) = 0, but also arcsin(0) = [[pi|{{pi}}]], arcsin(0) = 2{{pi}}, 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.
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| The principal inverses are listed in the following table.
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| {| class="wikitable" style="text-align:center"
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| |-
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| !Name
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| !Usual notation
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| !Definition
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| !Domain of ''x'' for real result
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| !Range of usual principal value <br /> ([[radian]]s)
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| !Range of usual principal value <br /> ([[Degree (angle)|degrees]])
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| |-
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| | '''arcsine''' || ''y'' = arcsin ''x'' || ''x'' = [[sine|sin]] ''y'' || −1 ≤ ''x'' ≤ 1 || −{{pi}}/2 ≤ ''y'' ≤ {{pi}}/2 || −90° ≤ ''y'' ≤ 90°
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| |-
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| | '''arccosine''' || ''y'' = arccos ''x'' || ''x'' = [[cosine|cos]] ''y'' || −1 ≤ ''x'' ≤ 1 || 0 ≤ ''y'' ≤ {{pi}} || 0° ≤ ''y'' ≤ 180°
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| |-
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| | '''arctangent''' || ''y'' = arctan ''x'' || ''x'' = [[trigonometric functions|tan]] ''y'' || all real numbers || −{{pi}}/2 < ''y'' < {{pi}}/2 || −90° < ''y'' < 90°
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| |-
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| | '''arccotangent''' || ''y'' = arccot ''x'' ||''x'' = [[cotangent|cot]] ''y'' || all real numbers
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| | 0 < ''y'' < {{pi}} || 0° < ''y'' < 180°
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| |-
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| | '''arcsecant''' || ''y'' = arcsec ''x'' || ''x'' = [[Trigonometric_functions#Reciprocal_functions|sec]] ''y'' || ''x'' ≤ −1 or 1 ≤ ''x'' || 0 ≤ ''y'' < {{pi}}/2 or {{pi}}/2 < ''y'' ≤ {{pi}} || 0° ≤ ''y'' < 90° or 90° < ''y'' ≤ 180°
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| |-
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| | '''arccosecant''' || ''y'' = arccsc ''x'' || ''x'' = [[cosecant|csc]] ''y'' || ''x'' ≤ −1 or 1 ≤ ''x'' || −{{pi}}/2 ≤ ''y'' < 0 or 0 < ''y'' ≤ {{pi}}/2 || -90° ≤ ''y'' < 0° or 0° < ''y'' ≤ 90°
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| |-
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| |}
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| If ''x'' is allowed to be a [[complex number]], then the range of ''y'' applies only to its real part.
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| ==Relationships among the inverse trigonometric functions==
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| [[Image:Arcsine Arccosine.svg|168px|right|thumb|The usual principal values of the arcsin(''x'') (red) and arccos(''x'') (blue) functions graphed on the cartesian plane.]]
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| [[Image:Arctangent Arccotangent.svg|294px|right|thumb|The usual principal values of the arctan(''x'') and arccot(''x'') functions graphed on the cartesian plane.]]
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| [[Image:Arcsecant Arccosecant.svg|294px|right|thumb|Principal values of the arcsec(''x'') and arccsc(''x'') functions graphed on the cartesian plane.]]
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| Complementary angles:
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| :<math>\arccos x = \frac{\pi}{2} - \arcsin x </math>
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| :<math>\arccot x = \frac{\pi}{2} - \arctan x </math>
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| :<math>\arccsc x = \frac{\pi}{2} - \arcsec x </math>
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| Negative arguments:
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| :<math>\arcsin (-x) = - \arcsin x \!</math>
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| :<math>\arccos (-x) = \pi - \arccos x \!</math>
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| :<math>\arctan (-x) = - \arctan x \!</math>
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| :<math>\arccot (-x) = \pi - \arccot x \!</math>
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| :<math>\arcsec (-x) = \pi - \arcsec x \!</math>
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| :<math>\arccsc (-x) = - \arccsc x \!</math>
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| Reciprocal arguments:
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| :<math>\arccos (1/x) \,= \arcsec x \,</math>
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| :<math>\arcsin (1/x) \,= \arccsc x \,</math>
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| :<math>\arctan (1/x) = \tfrac{1}{2}\pi - \arctan x =\arccot x,\text{ if }x > 0 \,</math>
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| :<math>\arctan (1/x) = -\tfrac{1}{2}\pi - \arctan x = -\pi + \arccot x,\text{ if }x < 0 \,</math>
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| :<math>\arccot (1/x) = \tfrac{1}{2}\pi - \arccot x =\arctan x,\text{ if }x > 0 \,</math>
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| :<math>\arccot (1/x) = \tfrac{3}{2}\pi - \arccot x = \pi + \arctan x,\text{ if }x < 0 \,</math>
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| :<math>\arcsec (1/x) = \arccos x \,</math>
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| :<math>\arccsc (1/x) = \arcsin x \,</math>
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| If you only have a fragment of a sine table:
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| :<math>\arccos x = \arcsin \sqrt{1-x^2},\text{ if }0 \leq x \leq 1 </math>
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| :<math>\arctan x = \arcsin \frac{x}{\sqrt{x^2+1}} </math>
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| 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).
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| From the [[tangent half-angle formula|half-angle formula]] <math>\tan \frac{\theta}{2} = \frac{\sin \theta}{1+\cos \theta} </math>, we get:
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| :<math>\arcsin x = 2 \arctan \frac{x}{1+\sqrt{1-x^2}}</math>
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| :<math>\arccos x = 2 \arctan \frac{\sqrt{1-x^2}}{1+x},\text{ if }-1 < x \leq +1 </math>
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| :<math>\arctan x = 2 \arctan \frac{x}{1+\sqrt{1+x^2}}</math>
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| ==Relationships between trigonometric functions and inverse trigonometric functions==
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| 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'' where ''x'' is any real number in the open interval {{open-open|''0'', ''1''}}, and possibly including either or both endpoints depending on whether the given is formula is defined for that ''x'' value, then applying the [[Pythagorean theorem]] and definitions the trigonometric ratios. Purely algebraic derivations are longer.
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| {|class="wikitable"
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| !<math> \theta </math>
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| !<math>\sin \theta </math>
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| !<math>\cos \theta </math>
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| !<math>\tan \theta </math>
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| !Diagram
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| !<math>\arcsin x </math>
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| |<math>\sin (\arcsin x) = x </math>
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| |<math>\cos (\arcsin x) = \sqrt{1-x^2}</math>
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| |<math>\tan (\arcsin x) = \frac{x}{\sqrt{1-x^2}}</math>
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| |[[file:Trigonometric functions and inverse3.svg|150px]]
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| |-
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| !<math>\arccos x </math>
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| |<math>\sin (\arccos x) = \sqrt{1-x^2}</math>
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| |<math>\cos (\arccos x) = x </math>
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| |<math>\tan (\arccos x) = \frac{\sqrt{1-x^2}}{x}</math>
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| |[[file:Trigonometric functions and inverse.svg|150px]]
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| |-
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| !<math>\arctan x </math>
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| |<math>\sin (\arctan x) = \frac{x}{\sqrt{1+x^2}}</math>
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| |<math>\cos (\arctan x) = \frac{1}{\sqrt{1+x^2}}</math>
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| |<math>\tan (\arctan x) = x</math>
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| |[[file:Trigonometric functions and inverse2.svg|150px]]
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| |-
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| !<math>\arccot x </math>
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| |<math>\sin (\arccot x) = \frac{1}{\sqrt{1+x^2}}</math>
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| |<math>\cos (\arccot x) = \frac{x}{\sqrt{1+x^2}}</math>
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| |<math>\tan (\arccot x) = \frac{1}{x}</math>
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| |[[file:Trigonometric functions and inverse4.svg|150px]]
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| !<math>\arcsec x </math>
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| |<math>\sin (\arcsec x) = \frac{\sqrt{x^2-1}}{x}</math>
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| |<math>\cos (\arcsec x) = \frac{1}{x}</math>
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| |<math>\tan (\arcsec x) = \sqrt{x^2-1}</math>
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| |[[file:Trigonometric functions and inverse6.svg|150px]]
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| !<math> \arccsc x </math>
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| |<math>\sin (\arccsc x) = \frac{1}{x}</math>
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| |<math>\cos (\arccsc x) = \frac{\sqrt{x^2-1}}{x}</math>
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| |<math>\tan (\arccsc x) = \frac{1}{\sqrt{x^2-1}}</math>
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| |[[file:Trigonometric functions and inverse5.svg|150px]]
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| |-
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| |}
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| ==General solutions==
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| 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'' − {{pi}}/2 (where ''k'' is an integer), finish it at 2{{pi}}''k'' + {{pi}}/2, and then reverse themselves over 2{{pi}}''k'' + {{pi}}/2 to 2{{pi}}''k'' + 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'' − {{pi}}/2, finishes it at 2{{pi}}''k'' + {{pi}}/2, and then repeats it (forward) over 2{{pi}}''k'' + {{pi}}/2 to 2{{pi}}''k'' + 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}}.
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| This periodicity is reflected in the general inverses where ''k'' is some integer:
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| :<math>\sin(y) = x \ \Leftrightarrow\ y = \arcsin(x) + 2k\pi \text{ or } y = \pi - \arcsin(x) + 2k\pi</math>
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| :Which, written in one equation, is: <math>\sin(y) = x \ \Leftrightarrow\ y = (-1)^k\arcsin(x) + k\pi</math>
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| :<math>\cos(y) = x \ \Leftrightarrow\ y = \arccos(x) + 2k\pi \text{ or } y = 2\pi - \arccos(x) + 2k\pi</math>
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| :Which, written in one equation, is: <math>\cos(y) = x \ \Leftrightarrow\ y = \pm\arccos(x) + 2k\pi</math>
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| :<math>\tan(y) = x \ \Leftrightarrow\ y = \arctan(x) + k\pi</math>
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| :<math>\cot(y) = x \ \Leftrightarrow\ y = \arccot(x) + k\pi</math>
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| :<math>\sec(y) = x \ \Leftrightarrow\ y = \arcsec(x) + 2k\pi \text{ or } y = 2\pi - \arcsec (x) + 2k\pi</math>
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| :<math>\csc(y) = x \ \Leftrightarrow\ y = \arccsc(x) + 2k\pi \text{ or } y = \pi - \arccsc(x) + 2k\pi</math>
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| == Extension to complex plane ==
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| Since the inverse trigonometric functions are [[analytic function]]s, they can be extended from the real line to the complex plane. This results in functions with multiple sheets and [[branch point]]s. One possible way of defining the extensions is:
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| :<math>\arctan z = \int_0^z \frac{d x}{1 + x^2} \quad z \neq -i, +i \,</math>
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| where the part of the imaginary axis which does not lie strictly between −''i'' and +''i'' is the cut between the principal sheet and other sheets;
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| :<math>\arcsin z = \arctan \frac{z}{\sqrt{1 - z^2}} \quad z \neq -1, +1 \,</math>
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| 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 cut between the principal sheet of arcsin and other sheets; | |
| :<math>\arccos z = \frac{\pi}{2} - \arcsin z \quad z \neq -1, +1 \,</math>
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| which has the same cut as arcsin;
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| :<math>\arccot z = \frac{\pi}{2} - \arctan z \quad z \neq -i, +i \,</math>
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| which has the same cut as arctan;
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| :<math>\arcsec z = \arccos \frac{1}{z} \quad z \neq -1, 0, +1 \,</math>
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| where the part of the real axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and other sheets;
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| :<math>\arccsc z = \arcsin \frac{1}{z} \quad z \neq -1, 0, +1 \,</math>
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| which has the same cut as arcsec.
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| ==Derivatives of inverse trigonometric functions==
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| :{{Main|Differentiation of trigonometric functions}}
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| The [[derivative]]s for complex values of ''z'' are as follows:
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| :<math>
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| \begin{align}
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| \frac{d}{dz} \arcsin z & {}= \frac{1}{\sqrt{1-z^2}}; \quad z \neq -1, +1 \\
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| \frac{d}{dz} \arccos z & {}= \frac{-1}{\sqrt{1-z^2}}; \quad z \neq -1, +1 \\
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| \frac{d}{dz} \arctan z & {}= \frac{1}{1+z^2}; \quad z \neq -i, +i \\
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| \frac{d}{dz} \arccot z & {}= \frac{-1}{1+z^2}; \quad z \neq -i, +i \\
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| \frac{d}{dz} \arcsec z & {}= \frac{1}{z^2\,\sqrt{1 - z^{-2}}}; \quad z \neq -1, 0, +1 \\
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| \frac{d}{dz} \arccsc z & {}= \frac{-1}{z^2\,\sqrt{1 - z^{-2}}}; \quad z \neq -1, 0, +1
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| \end{align}</math>
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| Only for real values of ''x'':
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| :<math>
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| \begin{align}
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| \frac{d}{dx} \arcsec x & {}= \frac{1}{|x|\,\sqrt{x^2-1}}; \qquad |x| > 1\\
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| \frac{d}{dx} \arccsc x & {}= \frac{-1}{|x|\,\sqrt{x^2-1}}; \qquad |x| > 1
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| \end{align}</math>
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| For a sample derivation: if <math>\theta = \arcsin x \!</math>, we get:
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| :<math>\frac{d \arcsin x}{dx} = \frac{d \theta}{d \sin \theta} = \frac{d \theta}{\cos \theta d \theta} = \frac{1} {\cos \theta} = \frac{1} {\sqrt{1-\sin^2 \theta}} = \frac{1}{\sqrt{1-x^2}}</math>
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| ==Expression as definite integrals==
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| Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral:
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| :<math>
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| \begin{align}
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| \arcsin x &{}= \int_0^x \frac {1} {\sqrt{1 - z^2}}\,dz,\qquad |x| \leq 1\\
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| \arccos x &{}= \int_x^1 \frac {1} {\sqrt{1 - z^2}}\,dz,\qquad |x| \leq 1\\
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| \arctan x &{}= \int_0^x \frac 1 {z^2 + 1}\,dz,\\
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| \arccot x &{}= \int_x^\infty \frac {1} {z^2 + 1}\,dz,\\
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| \arcsec x &{}= \int_1^x \frac 1 {z \sqrt{z^2 - 1}}\,dz, \qquad x \geq 1\\
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| \arcsec x &{}= \pi + \int_x^{-1} \frac 1 {z \sqrt{z^2 - 1}}\,dz, \qquad x \leq -1\\
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| \arccsc x &{}= \int_x^\infty \frac {1} {z \sqrt{z^2 - 1}}\,dz, \qquad x \geq 1\\
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| \arccsc x &{}= \int_{-\infty}^x \frac {1} {z \sqrt{z^2 - 1}}\,dz, \qquad x \leq -1
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| \end{align}</math>
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| When ''x'' equals 1, the integrals with limited domains are [[improper integral]]s, but still well-defined.
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| ==Infinite series==
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| Like the sine and cosine functions, the inverse trigonometric functions can be calculated using [[infinite series]], as follows:
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| :<math> \arcsin z = z + \left( \frac {1} {2} \right) \frac {z^3} {3}
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| + \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\
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| = \sum_{n=0}^\infty \frac {\binom{2n} n z^{2n+1}} {4^n (2n+1)}; \qquad | z | \le 1 </math>
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| <!-- extra blank line for legibility -->
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| :<math> \arccos z = \frac {\pi} {2} - \arcsin z
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| = \frac {\pi} {2} - \left( z + \left( \frac {1} {2} \right) \frac {z^3} {3} + \left( \frac {1 \cdot 3} {2 \cdot 4} \right) \frac {z^5} {5} + \cdots\ \right)
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| = \frac {\pi} {2} - \sum_{n=0}^\infty \frac {\binom{2n} n z^{2n+1}} {4^n (2n+1)}; \qquad | z | \le 1 </math>
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| <!-- extra blank line for legibility -->
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| :<math> \arctan z = z - \frac {z^3} {3} +\frac {z^5} {5} -\frac {z^7} {7} +\cdots\
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| = \sum_{n=0}^\infty \frac {(-1)^n z^{2n+1}} {2n+1}; \qquad | z | \le 1 \qquad z \neq i,-i </math>
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| <!-- extra blank line for legibility -->
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| :<math> \arccot z = \frac {\pi} {2} - \arctan z \ = \frac {\pi} {2} - \left( z - \frac {z^3} {3} +\frac {z^5} {5} -\frac {z^7} {7} +\cdots\ \right)
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| = \frac {\pi} {2} - \sum_{n=0}^\infty \frac {(-1)^n z^{2n+1}} {2n+1}; \qquad | z | \le 1 \qquad z \neq i,-i </math>
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| <!-- extra blank line for legibility -->
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| :<math> \arcsec z = \arccos {(1/z)}
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| = \frac {\pi} {2} - \left( z^{-1} + \left( \frac {1} {2} \right) \frac {z^{-3}} {3} + \left( \frac {1 \cdot 3} {2 \cdot 4} \right) \frac {z^{-5}} {5} + \cdots\ \right)
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| = \frac {\pi} {2} - \sum_{n=0}^\infty \frac {\binom{2n} n z^{-(2n+1)}} {4^n (2n+1)}; \qquad | z | \ge 1 </math>
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| <!-- extra blank line for legibility -->
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| :<math> \arccsc z = \arcsin {(1/z)}
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| = z^{-1} + \left( \frac {1} {2} \right) \frac {z^{-3}} {3} + \left( \frac {1 \cdot 3} {2 \cdot 4 } \right) \frac {z^{-5}} {5} +\cdots\
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| = \sum_{n=0}^\infty \frac {\binom{2n} n z^{-(2n+1)}} {4^n (2n+1)}; \qquad | z | \ge 1 </math>
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| <!-- extra blank line for legibility -->
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| [[Leonhard Euler]] found a more efficient series for the arctangent, which is:
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| :<math>\arctan z = \frac{z}{1+z^2} \sum_{n=0}^\infty \prod_{k=1}^n \frac{2k z^2}{(2k+1)(1+z^2)}.</math>
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| (Notice that the term in the sum for ''n'' = 0 is the [[empty product]] which is 1.)
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| <!-- extra blank line for legibility -->
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| Alternatively, this can be expressed:
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| :<math>\arctan z = \sum_{n=0}^\infty \frac{2^{\,2n}\,(n!)^2}{\left(2n+1\right)!} \; \frac{z^{\,2n+1}}{\left(1+z^2\right)^{n+1}}</math>
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| ==Continued fractions for arctangent==
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| Two alternatives to the power series for arctangent are these [[generalized continued fraction]]s:
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| :<math>
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| \arctan z=\cfrac{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}}}}}
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| =\cfrac{z} {1+\cfrac{(1z)^2} {3+\cfrac{(2z)^2} {5+\cfrac{(3z)^2} {7+\cfrac{(4z)^2} {9+\ddots\,}}}}}\,
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| </math>
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| 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'')<sup>2</sup>, with each perfect square appearing once. The first was developed by [[Leonhard Euler]]; the second by [[Carl Friedrich Gauss]] utilizing the [[Gaussian hypergeometric series]].
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| ==Indefinite integrals of inverse trigonometric functions==
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| For real and complex values of ''x'':
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| :<math>
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| \begin{align}
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| \int \arcsin x\,dx &{}= x\,\arcsin x + \sqrt{1-x^2} + C\\
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| \int \arccos x\,dx &{}= x\,\arccos x - \sqrt{1-x^2} + C\\
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| \int \arctan x\,dx &{}= x\,\arctan x - \frac{1}{2}\ln\left(1+x^2\right) + C\\
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| \int \arccot x\,dx &{}= x\,\arccot x + \frac{1}{2}\ln\left(1+x^2\right) + C\\
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| \int \arcsec x\,dx &{}= x\,\arcsec x - \ln\left[x\left(1+\sqrt{{x^2-1}\over x^2}\right)\right] + C\\
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| \int \arccsc x\,dx &{}= x\,\arccsc x + \ln\left[x\left(1+\sqrt{{x^2-1}\over x^2}\right)\right] + C
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| \end{align}</math>
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| For real ''x'' ≥ 1:
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| :<math>
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| \begin{align}
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| \int \arcsec x\,dx &{}= x\,\arcsec x - \ln\left(x+\sqrt{x^2-1}\right) + C\\
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| \int \arccsc x\,dx &{}= x\,\arccsc x + \ln\left(x+\sqrt{x^2-1}\right) + C
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| \end{align}</math>
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| All of these can be derived using [[integration by parts]] and the simple derivative forms shown above.
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| ===Example===
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| Using <math>\int u\,\mathrm{d}v = u v - \int v\,\mathrm{d}u</math>, set
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| :<math>
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| \begin{align}
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| u &{}=&\arcsin x &\quad\quad\mathrm{d}v = \mathrm{d}x\\
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| \mathrm{d}u &{}=&\frac{\mathrm{d}x}{\sqrt{1-x^2}}&\quad\quad{}v = x
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| \end{align}</math>
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| Then
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| :<math>\int \arcsin(x)\,\mathrm{d}x = x \arcsin x - \int \frac{x}{\sqrt{1-x^2}}\,\mathrm{d}x</math>
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| [[Integration by substitution|Substitute]]
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| : <math>k = 1 - x^2.\,</math>
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| Then
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| : <math>\mathrm{d}k = -2x\,\mathrm{d}x</math>
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| and
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| :<math>\int \frac{x}{\sqrt{1-x^2}}\,\mathrm{d}x = -\frac{1}{2}\int \frac{\mathrm{d}k}{\sqrt{k}} = -\sqrt{k}</math>
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| Back-substitute for ''x'' to yield
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| :<math>\int \arcsin(x)\, \mathrm{d}x = x \arcsin x + \sqrt{1-x^2}+C </math>
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| ==Two-argument variant of arctangent==
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| {{anchor|Two-argument variant of arctangent}}
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| {{main|atan2}}
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| The two-argument [[atan2]] function computes the arctangent of ''y'' / ''x'' given ''y'' and ''x'', but with a range of (−{{pi}}, {{pi}}]. 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.
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|
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| In terms of the standard arctan function, that is with range of (−{{pi}}/2, {{pi}}/2), it can be expressed as follows:
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| :<math>\operatorname{atan2}(y, x) = \begin{cases}
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| \arctan(\frac y x) & \qquad x > 0 \\
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| \pi + \arctan(\frac y x) & \qquad y \ge 0 , x < 0 \\
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| -\pi + \arctan(\frac y x) & \qquad y < 0 , x < 0 \\
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| \frac{\pi}{2} & \qquad y > 0 , x = 0 \\
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| -\frac{\pi}{2} & \qquad y < 0 , x = 0 \\
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| \text{undefined} & \qquad y = 0, x = 0
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| \end{cases}</math>
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| It also equals the [[principal value]] of the [[arg (mathematics)|arg]]ument of the [[complex number]] ''x'' + ''iy''.
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| This function may also be defined using the [[tangent half-angle formula]]e as follows:
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| :<math>\operatorname{atan2}(y, x)=2\arctan \frac{y}{\sqrt{x^2 + y^2} + x} </math>
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| 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.
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| The above argument order (''y'', ''x'') seems to be the most common, and in particular is used in [[ISO standard]]s such as the [[C (programming language)|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#Variations_and_notation|Atan2]].
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| ==Arctangent function with location parameter==
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| In many applications the solution <math>y</math> of the equation <math>x=\tan y</math> is to come as close as possible to a given value <math>-\infty<\eta<\infty</math>. The adequate solution is produced by the parameter modified arctangent function
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| :<math>y=\arctan_\eta x:=\arctan x+\pi\cdot\operatorname{rni}\frac{\eta-\arctan x}{\pi} \, .</math>
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| The function <math>\operatorname{rni}</math> rounds to the nearest integer.
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| ==Logarithmic forms==
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| These functions may also be expressed using [[complex logarithm]]s. This extends in a natural fashion their [[domain of a function|domain]] to the [[complex plane]].
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| :<math>
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| \begin{align}
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| \arcsin x &{}= -i\,\ln\left(i\,x+\sqrt{1-x^2}\right) &{}= \arccsc \frac{1}{x}\\[10pt]
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| \arccos x &{}= i\,\ln\left(x-i\,\sqrt{1-x^2}\right) = \frac{\pi}{2}\,+i\ln\left(i\,x+\sqrt{1-x^2}\right) = \frac{\pi}{2}-\arcsin x &{}= \arcsec \frac{1}{x}\\[10pt]
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| \arctan x &{}= \tfrac{1}{2}i\left(\ln\left(1-i\,x\right)-\ln\left(1+i\,x\right)\right) &{}= \arccot \frac{1}{x}\\[10pt]
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| \arccot x &{}= \tfrac{1}{2}i\left(\ln\left(1-\frac{i}{x}\right)-\ln\left(1+\frac{i}{x}\right)\right) &{}= \arctan \frac{1}{x}\\[10pt]
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| \arcsec x &{}= -i\,\ln\left(i\,\sqrt{1-\frac{1}{x^2}}+\frac{1}{x}\right) = i\,\ln\left(\sqrt{1-\frac{1}{x^2}}+\frac{i}{x}\right)+\frac{\pi}{2} = \frac{\pi}{2}-\arccsc x &{}= \arccos \frac{1}{x}\\[10pt]
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| \arccsc x &{}= -i\,\ln\left(\sqrt{1-\frac{1}{x^2}}+\frac{i}{x}\right) &{}= \arcsin \frac{1}{x}
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| \end{align}</math>
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| Elementary proofs of these relations proceed via expansion to exponential forms of the trigonometric functions.
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| ===Example proof===
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| :<math>\theta = \arcsin x </math>
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| :<math>\sin(\theta) = \sin(\arcsin x) </math>
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| :<math>\sin(\theta) = x </math>
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| Using the [[Trigonometric_function#Relationship_to_exponential_function_and_complex_numbers|exponential definition of sine]]
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| :<math>\frac{e^{i\phi} - e^{-i\phi}}{2i} = \sin(\phi) </math>
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| one obtains | |
| :<math>\frac{e^{i\theta} - e^{-i\theta}}{2i} = x </math> | |
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| Let
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| :<math>k=e^{i\,\theta}. \, </math> | |
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| Then
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| :<math>\frac{k-\frac{1}{k}}{2i} = x</math>
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| :<math>{k-\frac{1}{k}} = 2ix</math>
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| :<math>{k -2ix -\frac{1}{k}} = 0</math>
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| :<math>k^2-2\,i\,k\,x-1\,=\,0</math>
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| :<math>k = ix \pm \sqrt{1-x^2} \, </math>
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| :<math>e^{i\theta} = ix \pm \sqrt{1-x^2} \, </math>
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| :<math>i \theta = \ln \left(ix \pm \sqrt{1-x^2}\right) \, </math>
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| :<math>\theta = -i \ln \left(ix \pm \sqrt{1-x^2}\right) \, </math>
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| (the positive branch is chosen)
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| :<math>\theta = \arcsin x = -i \ln \left(ix + \sqrt{1-x^2}\right) \, </math>
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| ===Example proof (variant 2)===
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| :<math>\theta = \arcsin x </math>
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| :<math>e^{i\theta}= \cos (\theta) + i \sin(\theta)</math>
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| :Apply the natural logarithm, multiply by -i and substitute theta.
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| :<math>\arcsin x= -i \ln(\cos (\arcsin x) + i \sin(\arcsin x))</math>
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| :<math>\arcsin x= -i \ln(\sqrt{1-x^2} + i x)</math>
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| | |
| {| style="text-align:center"
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| |+ '''Inverse trigonometric functions in the [[complex plane]]'''
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| |[[Image:Complex arcsin.jpg|1000x140px|none]]
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| |[[Image:Complex arccos.jpg|1000x140px|none]]
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| |[[Image:Complex arctan.jpg|1000x140px|none]]
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| |[[Image:Complex ArcCot.jpg|1000x140px|none]]
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| |[[Image:Complex ArcSec.jpg|1000x140px|none]]
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| |[[Image:Complex ArcCsc.jpg|1000x140px|none]]
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| |-
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| |<math>
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| \arcsin(z)
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| </math>
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| |<math>
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| \arccos(z)
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| </math>
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| |<math>
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| \arctan(z)
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| </math>
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| |<math>
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| \arccot(z)
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| </math>
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| |<math>
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| \arcsec(z)
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| </math>
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| |<math>
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| \arccsc(z)
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| </math>
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| |}
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| ==Arctangent addition formula==
| |
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| :<math>\arctan u + \arctan v = \arctan \left( \frac{u+v}{1-uv} \right) \pmod \pi, \qquad u v \ne 1 \,.</math>
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| This is derived from the tangent [[Angle sum and difference identities|addition formula]]
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| :<math>\tan ( \alpha + \beta ) = \frac{\tan \alpha + \tan \beta} {1 - \tan \alpha \tan \beta} \,,</math>
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| | |
| by letting | |
| :<math>\alpha = \arctan u \,, \quad \beta = \arctan v \,.</math>
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| ==Application: finding the angle of a right triangle==
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| [[Image:Trigonometry triangle.svg|right|thumb|A right triangle.]]
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| 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, for example, it follows that
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| :<math>\theta = \arcsin \left( \frac{\text{opposite}}{\text{hypotenuse}} \right).</math>
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| Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the [[Pythagorean Theorem]]: <math>a^2+b^2=h^2</math> where <math>h</math> is the length of the hypotenuse. Arctangent comes in handy in this situation, as the length of the hypotenuse is not needed.
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| :<math>\theta = \arctan \left( \frac{\text{opposite}}{\text{adjacent}} \right).</math>
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| 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:
| |
| | |
| :<math>\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}.</math>
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| ==Practical considerations==
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| For angles near 0 and {{pi}}, arccosine is [[ill-conditioned]] and will thus calculate the angle with reduced accuracy in a computer implementation (due to the limited number of digits). Similarly, arcsine is inaccurate for angles near −{{pi}}/2 and {{pi}}/2. To achieve full accuracy for all angles, arctangent or [[atan2]] should be used for the implementation.
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| ==See also==
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| *[[Argument (complex analysis)]]
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| *[[Complex logarithm]]
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| *[[Gauss's continued fraction]]
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| *[[Inverse hyperbolic function]]
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| *[[List of integrals of inverse trigonometric functions]]
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| *[[List of trigonometric identities]]
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| *[[Square root]]
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| *[[Tangent half-angle formula]]
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| *[[Trigonometric function]]
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| ==References==
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| {{reflist}}
| |
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| ==External links==
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| * {{MathWorld | urlname=InverseTrigonometricFunctions | title=Inverse Trigonometric Functions}}
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| * {{MathWorld | urlname=InverseTangent | title=Inverse Tangent}}
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| {{DEFAULTSORT:Inverse Trigonometric Functions}}
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| [[Category:Trigonometry]]
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| [[Category:Elementary special functions]]
| |