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Some trigonometric identities Periodicity: sin x, cos x have period 2π; tan x, cot x have period π Symmetry: cos is an even function, sin, tan, cot are odd. Turning sin to cos: sin( π2 − x) = cos x cos( π2 − x) = sin x Relating tan and sec: 1 + tan2 x = sec2 x , 1 + cot2 x = csc2 x Expanding sums: sin(x ± y) = sin x cos y ± cos x sin y Products to sums: cos(x ± y) = cos x cos y ∓ sin x sin y , x+y x−y 1 sin + sin sin x cos y = 2 2 2 1 y+x y−x 1 y+x y−x cos x cos y = cos + cos , sin x sin y = cos − cos 2 2 2 2 2 2 Expand double angle: cos(2x) = cos2 x − sin2 x = 2 cos2 x − 1 = 1 − 2 sin2 x sin(2x) = 2 sin x cos x , or the other way around, go to double angle by sin x cos = 1 sin(2x) , 2 cos2 x = 1 + cos(2x) 2 , sin2 x = 1 − cos(2x) 2 Some integrals: Z sec x dx = ln | sec x + tan x| + C Z , csc x dx = − ln | csc x + cot x| + C Derivatives of inverse trigonometric functions: d 1 arcsin x = √ dx 1 − x2 , d 1 arccos x = − √ dx 1 − x2 , d 1 arctan x = dx 1 + x2 Hyperbolic functions Definitions: sinh x = ex − e−x 2 , cosh x = ex + e−x 2 Relations: d sinh x = cosh x , dx cosh2 x − sinh2 x = 1 , sinh−1 x = ln x + √ 1 + x2 , d cosh x = sinh x dx d 1 sinh−1 x = √ dx 1 + x2