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.

Inverse functions

The trigonometric functions are periodic, so we must restrict their domains before we are able to define a unique inverse. In the following, the functions on the left are defined by the equation on the right; these are not proved identities. The principal inverses are usually defined as:

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: egin{matrix}

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mbox{for} & -rac{pi}{2} le y le rac{pi}{2},

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& y = rcsin(x) & mbox{if and only if} & x = sin(y) \

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mbox{for} & 0 le y le pi,

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& y = rccos(x) & mbox{if and only if} & x = cos(y) \

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mbox{for} & -rac{pi}{2} < y < rac{pi}{2},

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& y = rctan(x) & mbox{if and only if} & x = an(y) \

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mbox{for} & -rac{pi}{2} le y le rac{pi}{2}, y e 0,

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& y = rccsc(x) & mbox{if and only if} & x = csc(y) \

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mbox{for} & 0 le y le pi, y e rac{pi}{2},

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& y = rcsec(x) & mbox{if and only if} & x = sec(y) \

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mbox{for} & -rac{pi}{2} < y < rac{pi}{2}, y e 0,

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& y = rccot(x) & mbox{if and only if} & x = cot(y) \

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end{matrix}

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For inverse trigonometric functions, the notations sin−1 and cos−1 are often used for arcsin and arccos, etc. When this notation is used, the inverse functions are sometimes confused with the multiplicative inverses of the functions. Our notation avoids such confusion.

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The following series definition may be obtained:

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:rcsin z = z + left( rac {1} {2} ight) rac {z^3} {3} + left( rac {1 cdot 3} {2 cdot 4} ight) rac {z^5} {5} + left( rac{1 cdot 3 cdot 5} {2 cdot 4 cdot 6 } ight) rac{z^7} {7} + cdots = sum_{n=0}^infty left( rac {(2n)!} {2^{2n}(n!)^2} ight) rac {z^{2n+1}} {(2n+1)} , left| z ight| < 1

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:rccos z = rac {pi} {2} - rcsin z = rac {pi} {2} - (z + left( rac {1} {2} ight) rac {z^3} {3} + left( rac {1 cdot 3} {2 cdot 4} ight) rac {z^5} {5} + left( rac{1 cdot 3 cdot 5} {2 cdot 4 cdot 6 } ight) rac{z^7} {7} + cdots ) = rac {pi} {2} - sum_{n=0}^infty left( rac {(2n)!} {2^{2n}(n!)^2} ight) rac {z^{2n+1}} {(2n+1)} , left| z ight| < 1

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:rctan z = z - rac {z^3} {3} +rac {z^5} {5} -rac {z^7} {7} +cdots = sum_{n=0}^infty rac {(-1)^n z^{2n+1}} {2n+1} , left| z ight| < 1

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:rccsc z = rcsinleft(z^{-1} ight) = z^{-1} + left( rac {1} {2} ight) rac {z^{-3}} {3} + left( rac {1 cdot 3} {2 cdot 4 } ight) rac {z^{-5}} {5} + left( rac {1 cdot 3 cdot 5} {2 cdot 4 cdot 6} ight) rac {z^{-7}} {7} +cdots = sum_{n=0}^infty left( rac {(2n)!} {2^{2n}(n!)^2} ight) rac {z^{-(2n+1)}} {2n+1} , left| z ight| > 1

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:rcsec z = rccosleft(z^{-1} ight) = rac {pi} {2} - (z^{-1} + left( rac {1} {2} ight) rac {z^{-3}} {3} + left( rac {1 cdot 3} {2 cdot 4} ight) rac {z^{-5}} {5} + left( rac{1 cdot 3 cdot 5} {2 cdot 4 cdot 6 } ight) rac{z^{-7}} {7} + cdots ) = rac {pi} {2} - sum_{n=0}^infty left( rac {(2n)!} {2^{2n}(n!)^2} ight) rac {z^{-(2n+1)}} {(2n+1)} , left| z ight| > 1

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:rccot z = rac {pi} {2} - rctan z = rac {pi} {2} - ( z - rac {z^3} {3} +rac {z^5} {5} -rac {z^7} {7} +cdots ) = rac {pi} {2} - sum_{n=0}^infty rac {(-1)^n z^{2n+1}} {2n+1} , left| z ight| < 1

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These functions may also be defined by proving that they are antiderivatives of other functions. Then each function is uniquely determined by its value at a single point:

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:rcsinleft(z ight) =

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int rac 1 {sqrt{1 - z^2}},dz + C

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:rccosleft(z ight) =

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int rac {-1} {sqrt{1 - z^2}},dz + C

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:rctanleft(z ight) =

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int rac 1 {1 + z^2}, dz + C

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:rccscleft(z ight) =

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int rac {-1} {z^2 sqrt{1 - rac{1}{z^2}} }, dz + C

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:,rcsecleft(z ight) =

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int rac 1 {|z| sqrt{z^2 - 1}}, dz + C

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:rccotleft(z ight) =

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int rac {-1} {z^2 + 1}, dz + C

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Note: arcsec can also mean arcsecond.

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