TRIGONOMETRY Right Triangle Definitions opp adj sin θ = cos θ = hyp hyp opp adj cotθ = tan θ = adj opp hyp hyp sec θ = csc θ = adj opp Circular Definitions sin θ = tan θ = sec θ = y cos θ = r y cotθ = x r csc θ = x Other Identities x r x tan x = y r sec x = sin x cot x = cos x 1 cos x sin x 1 csc x = cos x sin x y Reduction Formulas Sum and Difference Formulas Pythagorean Identities sin( − x ) = − sin( x ) cos(u ± v ) = cos u ⋅ cos v ∓ sin u ⋅ sin v sin(u ± v ) = sin u ⋅ cos v ± cos u ⋅ sin v tan u ± tan v tan(u ± v ) = 1 ∓ tan u ⋅ tan v sin θ + cos θ = 1 cos( − x ) = cos( x ) tan( − x ) = − tan( x ) cot( − x ) = − cot( x ) sec( − x ) = sec( x ) csc( − x ) = − csc( x ) Double Angle Formulas sin(2u ) = 2 sin u cos u 2 2 cos(2u ) = cos u − sin u Power Reducing Formulas 2 1 − cos(2u ) 2 2 1 + cos(2u ) 2 2 1 − cos(2u ) sin u = 2 cos(2u ) = 2 cos u − 1 cos u = 2 cos(2u ) = 1 − 2 sin u tan(2u ) = 2 tan u tan u = 2 1 − tan u cos u cos v = 0.5[cos(u − v ) + cos(u + v )] sin u cos v = 0.5[sin(u + v ) + sin(u − v )] 2 2 2 2 1 + tan θ = sec θ 1 + cot θ = csc θ ⎛π ⎞ ⎛π ⎞ − x ⎟ = cos x cos ⎜ − x ⎟ = sin x ⎝2 ⎠ ⎝2 ⎠ ⎛π ⎞ ⎛π ⎞ tan ⎜ − x ⎟ = cot x cot ⎜ − x ⎟ = tan x ⎝2 ⎠ ⎝2 ⎠ ⎛π ⎞ ⎛π ⎞ sec ⎜ − x ⎟ = csc x csc ⎜ − x ⎟ = sec x ⎝2 ⎠ ⎝2 ⎠ sin ⎜ Special Angles cos 0 = 1 cos π 3 = 6 sin 0 = 0 sin cos u sin v = 0.5[sin(u + v ) − sin(u − v )] 2 Cofunction Identities 1 + cos(2u ) Product to Sum Formulas sin u sin v = 0.5[cos(u − v ) − cos(u + v )] 2 π cos 2 = 6 1 2 π = 4 sin π = 4 2 2 2 cos π = 3 sin 2 π 1 cos 2 = 3 π =0 2 3 sin 2 π =1 2 Derivative Rules d dx d dx d dx sin x = cos x 2 tan x = sec x sec x = sec x tan x d dx d dx d dx cos x = − sin x 2 cot x = − csc x csc x = − csc x cot x d dx d dx d dx arcsin x = arctan x = arc sec x = 1 d 1− x 1 dx d 2 2 x +1 1 2 x x −1 dx d dx ln x = 1 x cosh x = sinh x sinh x = cosh x d x x a = ln a ⋅ a dx d n n −1 x = n⋅x dx