Inverse Trigonometric Functions 2

8/13/2019 Inverse Trigonometric Functions 2

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Inverse trigonometric functions

Expression as definite integrals [edit ]

Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definiteintegral:

When x equals 1, the integrals with limited domains are improper integrals, but still well-defined.
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Infinite series [edit ]

Like the sine and cosine functions, the inverse trigonometric functions can be calculated using infinite series, as follows:
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Leonhard Euler found a more efficient series for the arctangent, which is:

(Notice that the term in the sum for n = 0 is the empty product which is 1.)

Alternatively, this can be expressed:

Continued fractions for arctangent [edit ]

Two alternatives to the power series for arctangent are these generalized continued fractions:
http://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Empty_producthttp://en.wikipedia.org/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=11http://en.wikipedia.org/wiki/Generalized_continued_fractionhttp://en.wikipedia.org/wiki/Generalized_continued_fractionhttp://en.wikipedia.org/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=11http://en.wikipedia.org/wiki/Empty_producthttp://en.wikipedia.org/wiki/Leonhard_Euler


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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 imaginaryaxis, 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 partialdenominators are the odd natural numbers, and the partial numerators (after the first) are just ( nz ) 2, with each perfect square appearingonce. The first was developed by Leonhard Euler; the second by Carl Friedrich Gauss utilizing the Gaussian hypergeometric series.

Indefinite integrals of inverse trigonometric functions [ edit ]

For real and complex values of x:

For real x 1:
http://en.wikipedia.org/wiki/Leonhard_Eulerhttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Gaussian_hypergeometric_serieshttp://en.wikipedia.org/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=12http://en.wikipedia.org/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=12http://en.wikipedia.org/wiki/Gaussian_hypergeometric_serieshttp://en.wikipedia.org/wiki/Carl_Friedrich_Gausshttp://en.wikipedia.org/wiki/Leonhard_Euler


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

Example [edit]

Using set

Then

Substitute

Then

And
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Back-substitute for x to yield

Two-argument variant of arctangent [edit ]

Main article: atan2

The two-argument atan2 function computes the arctangent of y / x given y and x, but with a range of ( , ]. 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 ( /2, /2), it can be expressed as follows:

It also equals the principal value of the argument of the complex number x + iy.
http://en.wikipedia.org/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=14http://en.wikipedia.org/wiki/Atan2http://en.wikipedia.org/wiki/Atan2http://en.wikipedia.org/wiki/Principal_valuehttp://en.wikipedia.org/wiki/Arg_(mathematics)http://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Complex_numberhttp://en.wikipedia.org/wiki/Arg_(mathematics)http://en.wikipedia.org/wiki/Principal_valuehttp://en.wikipedia.org/wiki/Atan2http://en.wikipedia.org/wiki/Atan2http://en.wikipedia.org/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=14


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This function may also be defined using the tangent half-angle formulae as follows:

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 programminglanguage, but a few authors may use the opposite convention ( x, y) so some caution is warranted. These variations are detailed atAtan2.

Arctangent function with location parameter [ edit ]

In many applications the solution of the equation is to come as close as possible to a given value . Theadequate solution is produced by the parameter modified arctangent function

The function rounds to the nearest integer.

Logarithmic forms [ edit ]

These functions may also be expressed using complex logarithms. This extends in a natural fashion their domain to the complex plane.
http://en.wikipedia.org/wiki/Tangent_half-angle_formulahttp://en.wikipedia.org/wiki/ISO_standardhttp://en.wikipedia.org/wiki/C_(programming_language)http://en.wikipedia.org/wiki/C_(programming_language)http://en.wikipedia.org/wiki/Atan2#Variations_and_notationhttp://en.wikipedia.org/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=15http://en.wikipedia.org/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=16http://en.wikipedia.org/wiki/Complex_logarithmhttp://en.wikipedia.org/wiki/Domain_of_a_functionhttp://en.wikipedia.org/wiki/Complex_planehttp://en.wikipedia.org/wiki/Complex_planehttp://en.wikipedia.org/wiki/Domain_of_a_functionhttp://en.wikipedia.org/wiki/Complex_logarithmhttp://en.wikipedia.org/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=16http://en.wikipedia.org/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=15http://en.wikipedia.org/wiki/Atan2#Variations_and_notationhttp://en.wikipedia.org/wiki/C_(programming_language)http://en.wikipedia.org/wiki/C_(programming_language)http://en.wikipedia.org/wiki/ISO_standardhttp://en.wikipedia.org/wiki/Tangent_half-angle_formula


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Elementary proofs of these relations proceed via expansion to exponential forms of the trigonometric functions.

Example proof [edit]

Using the exponential definition of sine
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One obtains

Let

Then

(The positive branch is chosen)


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Example proof (variant 2) [edit]

Apply the natural logarithm, multiply by -i and substitute theta.

Inverse trigonometric functions in the complex plane

Arctangent addition formula [ edit ]

This is derived from the tangent addition formula

By letting
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e:Complex_arcsin.jpghttp://en.wikipedia.org/wiki/File:Complex_ArcCsc.jpghttp://en.wikipedia.org/wiki/File:Complex_ArcSec.jpghttp://en.wikipedia.org/wiki/File:Complex_ArcCot.jpghttp://en.wikipedia.org/wiki/File:Complex_arctan.jpghttp://en.wikipedia.org/wiki/File:Complex_arccos.jpghttp://en.wikipedia.org/wiki/File:Complex_arcsin.jpghttp://en.wikipedia.org/wiki/File:Complex_ArcCsc.jpghttp://en.wikipedia.org/wiki/File:Complex_ArcSec.jpghttp://en.wikipedia.org/wiki/File:Complex_ArcCot.jpghttp://en.wikipedia.org/wiki/File:Complex_arctan.jpghttp://en.wikipedia.org/wiki/File:Complex_arccos.jpghttp://en.wikipedia.org/wiki/File:Complex_arcsin.jpghttp://en.wikipedia.org/wiki/File:Complex_ArcCsc.jpghttp://en.wikipedia.org/wiki/File:Complex_ArcSec.jpghttp://en.wikipedia.org/wiki/File:Complex_ArcCot.jpghttp://en.wikipedia.org/wiki/File:Complex_arctan.jpghttp://en.wikipedia.org/wiki/File:Complex_arccos.jpghttp://en.wikipedia.org/wiki/File:Complex_arcsin.jpghttp://en.wikipedia.org/wiki/Angle_sum_and_difference_identitieshttp://en.wikipedia.org/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=19http://en.wikipedia.org/wiki/Complex_planehttp://en.wikipedia.org/w/index.php?title=Inverse_trigonometric_functions&action=edit&section=18


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Application: finding the angle of a right triangle [edit ]

A right triangle.

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

Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the Pythagorean Theorem: where 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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For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle with the horizontal, where may becomputed as follows:

Practical considerations [edit ]

For angles near 0 and , arccosine is ill-conditioned and will thus calculate the angle with reduced accuracy in a computerimplementation (due to the limited number of digits). Similarly, arcsine is inaccurate for angles near /2 and /2. To achieve fullaccuracy for all angles, arctangent or atan2 should be used for the implementation.
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Inverse Trigonometric Functions 2

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