Trigonometric Formulas
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Trigonometric Formulas Aaron Peterson 1 2 Definitions 1. tan(x) = sin(x) cos(x) 2. sec(x) = 1 cos(x) 3. csc(x) = 1 sin(x) 4. cot(x) = 1 tan(x) = cos(x) sin(x) Pythagorean 1. sin2 (x) + cos2 (x) = 1 2. 1 + cot2 (x) = csc2 (x) 3. tan2 (x) + 1 = sec2 (x) 3 Angle Sum and Difference 1. sin(φ ± θ) = sin(φ) cos(θ) ± cos(φ) sin(θ) 2. cos(φ ± θ) = cos(φ) cos(θ) ∓ sin(φ) sin(θ) 3. tan(φ ± θ) = 4 5 tan(φ)±tan(θ) 1∓tan(φ) tan(θ) Combination With Different Arguments 1. sin(mx) sin(nx) = 1 2 (cos([m − n]x) − cos([m + n]x)) 2. sin(mx) cos(nx) = 1 2 (sin([m − n]x) + sin([m + n]x)) 3. cos(mx) cos(nx) = 1 2 (cos([m − n]x) + cos([m + n]x)) Power-Reducing 1. sin2 (θ) = 1−cos(2θ) 2 2. cos2 (θ) = 1+cos(2θ) 2 1 6 Half-Angle q 1. sin( θ2 ) = ± 1−cos(θ) 2 q 2. cos( θ2 ) = ± 1+cos(θ) 2 3. tan( θ2 ) = csc(θ) − cot(θ) 7 Multiple Angle Formulae 1. sin(2θ) = 2 sin(θ) cos(θ) 2. cos(2θ) = cos2 (θ) − sin2 (θ) 2 tan(θ) 1−tan2 (θ) 3. tan(2θ) = 4. sin(nθ) = Pn n k 5. cos(nθ) = Pn n k Where n k k=0 = k=0 cosk (θ) sinn−k (θ) sin cosk (θ) sinn−k (θ) cos 1 2 (n 1 2 (n − k)π − k)π , n! k! (n−k)! . 2
