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.

Identities

:sin left(x+y ight)=sin x cos y + cos x sin y

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:sin left(x-y ight)=sin x cos y - cos x sin y

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:cos left(x+y ight)=cos x cos y - sin x sin y

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:cos left(x-y ight)=cos x cos y + sin x sin y

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:sin x+sin y=2sin left( rac{x+y}{2} ight) cos left( rac{x-y}{2} ight)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:sin x-sin y=2cos left( rac{x+y}{2} ight) sin left( rac{x-y}{2} ight)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:cos x+cos y=2cos left( rac{x+y}{2} ight) cos left( rac{x-y}{2} ight)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:cos x-cos y=-2sin left( rac{x+y}{2} ight)sin left( rac{x-y}{2} ight)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: an x+ an y=rac{sin left( x+y ight) }{cos xcos y}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

: an x- an y=rac{sin left( x-y ight) }{cos xcos y}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:cot x+cot y=rac{sin left( x+y ight) }{sin xsin y}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:cot x-cot y=rac{-sin left( x-y ight) }{sin xsin y}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

See also trigonometric identity.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~