Trigonometric Identities

Pronunciation: /ˌtrɪg ə nəˈmɛ trɪk ɪˈdɛn tɪ tiz/ ?

A trigonometric identity is a trigonometric equation that is always true. Trigonometric identities are used to solve problems involving trigonometric functions. They are particularly useful for calculus.

The common trigonometric identities are:

Pythagorean Identities

cos(theta)^2+sin(theta)^2=1 (derivation)
tan(theta)^2+1=sec(theta)^2 (derivation)
cot(theta)^2+1=csc(theta)^2 (derivation)

Co-function Identities

sin(pi/2-u)=cos(u) (derivation)
cos(pi/2-u)=sin(u) (derivation)
tan(pi/2-u)=cot(u) (derivation)
cot(pi/2-u)=tan(u) (derivation)
csc(pi/2-u)=sec(u) (derivation)
sec(pi/2-u)=csc(u) (derivation)

Even Odd Identities

sin(-u)=-sin(u)
cos(-u)=cos(u)
tan(-u)=-tan(u)
csc(-u)=-csc(u)
sec(-u)=sec(u)
cot(-u)=-cot(u)

Sum Difference Identities

sin(u+-v)=sin(u)cos(v)+-cos(u)sin(v)
cos(u+-v)=cos(u)cos(v)-+sin(u)sin(v)
tan(u+-v)=(tan(u)+-tan(v))/(1-+tan(u)tan(v)

Double Angle Identities

sin(2u)=2sin(u)cos(u)
cos(2u)=cos^2(u)-sin^2(u)
cos(2u)=2cos^2(u)-1
cos(2u)=1-2sin^2(u)
tan(2u)=(2tan(u))/(1-tan^2(u))

Half Angle Identities

sin^2(u)=(1-cos(2u))/2
cos^2(u)=(1+cos(2u))/2
tan^2(u)=(1-cos(2u)/(1+cos(2u))

Sum to Product Identities

sin(u)+sin(v)=2sin((u+v)/2)cos((u-v)/2)
sin(u)-sin(v)=2cos((u+v)/2)sin((u-v)/2)
cos(u)+cos(v)=2cos((u+v)/2)cos((u-v)/2)
cos(u)-cos(v)=-2sin((u+v)/2)sin((u-v)/2)

Product to Sum Identities

sin(u)sin(v)=(1/2)(cos(u-v)-cos(u+v))
cos(u)cos(v)=(1/2)(cos(u-v)+cos(u+v))
sin(u)cos(v)=(1/2)(sin(u+v)+sin(u-v))
cos(u)sin(v)=(1/2)(sin(u+v)-sin(u-v))
Table 1: Trigonometric identities

Cite this article as:


Trigonometric Identities. 2011-01-03. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/t/trigonometricidentities.html.

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2011-01-03: Added external links to derivation of Pythagorean identities and cofunction identities. (McAdams, David.)
2008-07-01: Initial version (McAdams, David.)

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