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MTH 132 Lecture 9 Trigonometric Derivatives Sine and Cosine addition ● sin(a+b) = sin(a)cos(b)+cos(a)sin(b) ● cos(a+b) = cos(a)cos(b)sin(a)sin(b) Derivative of Sine ● sin(x)’ = cos(x) Sine Proof ● [ sin(x+h) sin(x) ] / h ● [ Sin(x)cos(h) + cos(x)sin(h) sin(x) ] / h ● Cos(x) * sin(h)/h sin(x) * (cos(h) 1)/h ○ [cos(h) 1]/h always = 0 ● Simplify to get cos x as h approaches 0. Cosine Derivative ● cos(x)’ = sinx Cosine Proof ● f(x) = cosx ● f’(x) = limit h approaches 0 [ f(x+h) f(x) ]/ h ● cos(x+h) cos(x) = cos(x)cos(h) sin(x)sin(h) cos(x) ● cos(x)*[cos(h)1]/h sinx*sin(h)/h ● sin^2x+cos^2x = 1 ● (sin^2x+cos^2x )’= 1’ ● (sinx*sinx)’ + (cosx*