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# Trigonometry

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TRIGONOMETRIC IDENTITIES • 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 π π sin( − u) = cos u cos( − u) = sin u 2 2 tan( csc( π π − u) = cot u cot( − u) = tan u 2 2 sec( π − u) = csc u 2 • Sum-to-Product Formulas sin u + sin v = 2 sin u+v 2 u+v 2 u+v 2 u+v 2 cos u−v 2 u−v 2 u−v 2 u−v 2 • Power-Reducing/Half Angle Formulas sin2 u = 1 − cos(2u) 2 1 + cos(2u) cos2 u = 2 1 − cos(2u) tan2 u = 1 + cos(2u)

sin u − sin v = 2 cos

sin

cos u + cos v = 2 cos

cos

cos u − cos v = −2 sin

sin

π − u) = sec u 2

• Product-to-Sum Formulas sin u sin v = cos u cos v = sin u cos v = cos u sin v = 1 [cos(u − v) − cos(u + v)] 2 1 [cos(u − v) + cos(u + v)] 2 1 [sin(u + v) + sin(u − v)] 2 1 [sin(u + v) − sin(u − v)] 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

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