List of Trigonometric Identities
Knowledge
Trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined.
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions.
Trigonometry Function Identities
Co-function Identities
sinθ = cos(π/2 - θ)
secθ = csc(π/2 - θ)
tanθ = cot(π/2 - θ)
Negative Angle Identities
sin(-θ) = - sinθ
cos(-θ) = cosθ
tan(-θ) = - tanθ
csc(-θ) = - cscθ
sec(-θ) = secθ
cot(-θ) = -cotθ
Addition and Subtraction Identities
sin( A + B ) = sinA cosB + cosA sinB
cos( A + B ) = cosA cosB - sinA sinB
tan( A + B ) = = tanA + tanB1 - tanA tanB
sin( A - B ) = sinA cosB - cosA sinB
cos( A - B ) = cosA cosB - sinA sinB
tan( A - B ) = = tanA - tanB1 + tanA tanB
Double-Angle Identities
sin2θ = 2 sinθ cosθ
cos2θ = cos2θ - sin2θ
cos2θ = 2cos2θ - 1
cos2θ = 1 - 2sin2θ
tan2θ = 2tanθ1-tan2θ
Product Identities
sinA cosB = 12( sin(A+B) + sin(A-B) )
cosA sinB = 12( sin(A+B) - sin(A-B) )
cosA cosB = 12( cos(A+B) + cos(A-B) )
sinA sinB = 12( cos(A-B) - cos(A+B) )
Supplement Angle Identities
sin( π - θ ) = sinθ
cos( π - θ ) = - cosθ
tan( π - θ ) = - tanθ
sin( π + θ ) = - sinθ
cos( π + θ ) = - cosθ
tan( π + θ ) = tanθ
csc( π - θ ) = cscθ
sec( π - θ ) = - secθ
cot( π - θ ) = - cotθ
csc( π + θ ) = - cscθ
sec( π + θ ) = - secθ
cot( π + θ ) = cotθ
Quotient Identities
tanθ = sinθcosθ
secθ = 1cosθ
cotθ = cosθsinθ = 1tanθ
cscθ = 1sinθ
Pythagorean Identities
sin2θ + cos2θ = 1
tan2θ + 1 = sec2θ
cot2θ + 1 = csc2θ
Half-Angle Identities
sinθ2 = ± 1-cosθ2
cosθ2 = ± 1+cosθ2
tanθ2 = ± 1-cosθ1+cosθ
Sum Identities
sinA + sinB = 2sin(A+B2)cos(A-B2)
sinA - sinB = 2cos(A+B2)sin(A-B2)
cosA + cosB = 2cos(A+B2)cos(A-B2)
cosA - cosB = -2cos(A+B2)sin(A-B2)
Example
θ1 is located in Quadrant III, and cos(θ1) = - 35
sin2(θ1) + cos2(θ1) = 1
sin2(θ1) + (- 35 )2 = 1
sin2(θ1) = 1 - (- 35 )2
sin(θ1) = ± 45
The sign of sin(θ1) is determined by the quadrant. θ1 is in Quadrant Ⅲ, so its sine value must be negative. In conclusion, sin(θ1) = - 45 .
