Trigonometric function

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to positive and negative values and even to complex numbers. All of these approaches will be presented below.

Series definitions

Please note: Here, and generally in calculus, all angles are measured in radians. (See also below).

Related Topics:
Calculus - Radian

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Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine and the derivative of cosine is negative sine. One can then use the theory of Taylor series to show that the following identities hold for all real numbers x:

Related Topics:
Limits - Derivative - Taylor series - Real number

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:sin x = x - rac{x^3}{3!} + rac{x^5}{5!} - rac{x^7}{7!} + cdots = sum_{n=0}^infty rac{(-1)^nx^{2n+1}}{(2n+1)!}

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:cos x = 1 - rac{x^2}{2!} + rac{x^4}{4!} - rac{x^6}{6!} + cdots = sum_{n=0}^infty rac{(-1)^nx^{2n}}{(2n)!}

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These identities are often taken as the definitions of the sine and cosine function. They are often used as the starting point in a rigorous treatment of trigonometric functions and their applications (e.g. in Fourier series), since the theory of infinite series can be developed from the foundations of the real number system, independent of any geometric considerations. The differentiability and continuity of these functions are then established from the series definitions alone.

Related Topics:
Fourier series - Infinite series - Real number system - Differentiability - Continuity

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Other series can be found:

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: an x = x + rac{x^3}{3} + rac{2 x^5}{15} + rac{17 x^7}{315} + cdots = sum_{n=1}^infty rac{2^{2n} (2^{2n}-1) B_n x^{2n-1}}{(2n)!}, left | x ight | < rac {pi} {2}

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:csc x = rac {1} {x} + rac {x} {6} + rac {7 x^3} {360} + rac {31 x^5} {15120} + cdots = rac {1} {x} + sum_{n=1}^infty rac{2 (2^{2n-1}-1) B_n x^{2n-1}}{(2n)!}, 0 < left | x ight | < rac {pi} {2}

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:sec x = 1 + rac {x^2} {2} + rac {5 x^4} {24} + rac {61 x^6} {720} + cdots = 1+ sum_{n=1}^infty rac{E_n x^{2n}}{(2n)!}, left | x ight | < rac {pi} {2}

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:cot x = rac {1} {x} - rac {x}{3} - rac {x^3} {45} - rac {2 x^5} {945} - cdots = rac {1} {x} - sum_{n=1}^infty rac{2^{2n} B_n x^{2n-1}}{(2n)!}, 0 < left | x ight | < rac {pi} {2}

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where

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:B_n , is the nth Boustrophedon Transform number

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:E_n , is the nth Euler number

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Relationship to exponential function

It can be shown from the series definitions that the sine and cosine functions are the imaginary and real parts, respectively, of the complex exponential function when its argument is purely imaginary:

Related Topics:
Imaginary - Complex exponential function

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: e^{i heta} = cos heta + isin heta ,.

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This relationship was first noted by Euler and the identity is called Euler's formula. In this way, trigonometric functions become essential in the geometric interpretation of complex analysis. For example, with the above identity, if one considers the unit circle in the complex plane, defined by eix, and as above, we can parametrize this circle in terms of cosines and sines, the relationship between the complex exponential and the trigonometric functions becomes more apparent.

Related Topics:
Euler - Euler's formula - Complex analysis - Complex plane

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Furthermore, this allows for the definition of the trigonometric functions for complex arguments z:

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: sin z , = , sum_{n=0}^{infty}rac{(-1)^{n}}{(2n+1)!}z^{2n+1} , = , {e^{imath z} - e^{-imath z} over 2imath} = -imath sinh left( imath z ight)

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: cos z , = , sum_{n=0}^{infty}rac{(-1)^{n}}{(2n)!}z^{2n} , = , {e^{imath z} + e^{-imath z} over 2} = cosh left(imath z ight)

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where i2 = −1. Also, for purely real x,

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: cos x , = , mbox{Re } (e^{imath x})

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: sin x , = , mbox{Im } (e^{imath x})

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It is also shown that exponential processes are intimately linked to periodic behavior.

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