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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

= 2cos2θ - 1

= 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

sin21) + cos21) = 1

sin21) + (- 35 )2 = 1

sin21) = 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 .

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