TRIGONOMETRIC IDENTITIES Reciprocal identities Power ...

TRIGONOMETRIC IDENTITIES

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? Reciprocal identities

1

1

sin u =

cos u =

csc u

sec u

1

1

tan u =

cot u =

cot u

tan u

1

1

csc u =

sec u =

sin u

cos u

? Pythagorean Identities

sin2 u + cos2 u = 1

1 + tan2 u = sec2 u 1 + cot2 u = csc2 u

? Quotient Identities

sin u

cos u

tan u =

cot u =

cos u

sin u

? Co-Function Identities

sin( - u) = cos u cos( - u) = sin u

2

2

tan( - u) = cot u cot( - u) = tan u

2

2

csc( - u) = sec u sec( - u) = csc u

2

2

? Parity Identities (Even & Odd)

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

? Sum & Difference Formulas sin(u ? v) = sin u cos v ? cos u sin v cos(u ? v) = cos u cos v sin u sin v tan u ? tan v tan(u ? v) = 1 tan u tan v

? Double Angle Formulas

sin(2u) = 2 sin u cos u cos(2u) = cos2 u - sin2 u

= 2 cos2 u - 1 = 1 - 2 sin2 u

2 tan u tan(2u) = 1 - tan2 u

? Power-Reducing/Half Angle Formulas

sin2 u = 1 - cos(2u) 2

cos2 u = 1 + cos(2u) 2

tan2 u = 1 - cos(2u) 1 + cos(2u)

? Sum-to-Product Formulas

u+v

u-v

sin u + sin v = 2 sin

cos

2

2

u+v

u-v

sin u - sin v = 2 cos

sin

2

2

u+v

u-v

cos u + cos v = 2 cos

cos

2

2

u+v

u-v

cos u - cos v = -2 sin

sin

2

2

? Product-to-Sum Formulas 1

sin u sin v = [cos(u - v) - cos(u + v)] 2 1

cos u cos v = [cos(u - v) + cos(u + v)] 2 1

sin u cos v = [sin(u + v) + sin(u - v)] 2 1

cos u sin v = [sin(u + v) - sin(u - v)] 2

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