ReeTutorsMath.org - Trigonometry Formulas - Page 1 of 2 Double-Angle Formulas Definitions (OPPosite, ADJacent, HYPotenouse) sin(2θ) = 2 sin(θ) cos(θ) opp sin(θ) = hyp hyp csc(θ) = opp cos(2θ) = cos2 (θ) − sin2 (θ) cos(2θ) = 2 cos2 (θ) − 1 cos(θ) = adj hyp opp tan(θ) = adj sec(θ) = hyp adj adj cot(θ) = opp Definitions - (x, y, r) sin(θ) = y r csc(θ) = r y cos(θ) = x r sec(θ) = r x y x cot(θ) = tan(θ) = x y Changing To Sine And Cosine tan(θ) = sin(θ) cos(θ) sec(θ) = 1 cos(θ) cot(θ) = cos(θ) sin(θ) csc(θ) = 1 sin(θ) cos(2θ) = 1 − 2 sin2 (θ) tan(2θ) = 2 tan(θ) 1 − tan2 (θ) Half-Angle Formulas θ = sin 2 θ cos = 2 r 1 − cos(θ) 2 r 1 + cos(θ) 2 + − + − θ 1 − cos(θ) tan = 2 sin(θ) θ sin(θ) tan = 2 1 + cos(θ) Sum and Difference Formulas sin(A + B) = sin(A) cos(B) + cos(A) sin(B) Pythagorean Identities sin2 (θ) + cos2 (θ) = 1 sin(A − B) = sin(A) cos(B) − cos(A) sin(B) cos(A + B) = cos(A) cos(B) − sin(A) sin(B) cos(A − B) = cos(A) cos(B) + sin(A) sin(B) tan2 (θ) + 1 = sec2 (θ) 1 + cot2 (θ) = csc2 (θ) p sin(θ) = + 1 − cos2 (θ) − p cos(θ) = + 1 − sin2 (θ) − tan(A + B) = tan(A) + tan(B) 1 − tan(A) tan(B) tan(A − B) = tan(A) − tan(B) 1 + tan(A) tan(B) www.ReeTutorsMath.org/notes ReeTutorsMath.org - Trigonometry Formulas - Page 2 of 2 Cofunction Formulas (in degrees) Product-to-Sum Formulas sin(A) sin(B) = 1 [cos(A − B) − cos(A + B)] 2 cos(A) cos(B) = 1 [cos(A − B) + cos(A + B)] 2 sin(A) cos(B) = 1 [sin(A − B) + sin(A + B)] 2 sin(θ) cos(θ) tan(θ) cot(θ) sec(θ) csc(θ) = = = = = = cos(90◦ − θ) sin(90◦ − θ) cot(90◦ − θ) tan(90◦ − θ) csc(90◦ − θ) sec(90◦ − θ) Cofunction Formulas (in radians) Sum-to-Product Formulas sin(A) + sin(B) = 2 sin A+B 2 sin(A) − sin(B) = 2 cos cos(A) + cos(B) = 2 cos cos A+B 2 A+B 2 cos(A) − cos(B) = −2 sin sin A+B 2 sin(θ) = cos A−B 2 A−B 2 cos A−B 2 sin π cos(θ) = sin 2 π tan(θ) = cot A−B 2 cot(θ) = tan sec(θ) = csc csc(θ) = sec 2 π 2 π 2 π 2 π 2 −θ −θ −θ −θ −θ −θ Power Reducing Formulas sin2 (θ) = Derivative Formulas (Calculus) 1 − cos(2θ) 2 1 + cos(2θ) cos2 (θ) = 2 1 − cos(2θ) tan2 (θ) = 1 + cos(2θ) Even and Odd Trig Functions sin(−θ) csc(−θ) tan(−θ) cot(−θ) cos(−θ) sec(−θ) = = = = = = − sin(θ) − csc(θ) − tan(θ) − cot(θ) cos(θ) sec(θ) sine is ODD cosecant is ODD tangent is ODD cotangent is ODD cosine is EVEN secant is EVEN d sin(x) dx d cos(x) dx d tan(x) dx d cot(x) dx d sec(x) dx d csc(x) dx www.ReeTutorsMath.org/notes = cos(x) = − sin(x) = sec2 (x) = − csc2 (x) = sec(x) tan(x) = − csc(x) cot(x)