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lec10 - 9 Derivatives of Trig Functions P K Lamm Lecture...

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9. Derivatives of Trig Functions P. K. Lamm 10/02/11 (15:33) p. 1 / 5 Lecture Notes: 9. Derivatives of Trig Functions These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. The trigonometric limits lim x 0 sin x x = 1 , and lim x 0 cos x - 1 x = 0 , are needed in order to derive the derivative of trig functions. Recall that these limits were valid provided that angles are measured in radians ; because of this, the derivatives found below of trig functions also assume that angles are measured in radians. 1. Derivatives of the Sine and Cosine Functions Both of the trig limits are needed to find the derivatives of sin x and cos x : Theorem: (sin x ) 0 = cos x . Proof: Let f ( x ) = sin x and apply the usual definition of the derivative of f at x . That is, f 0 ( x ) = lim h 0 sin( x + h ) - sin x h . Using the trig identity sin( A + B ) = sin A cos B + cos A sin B, it follows that f 0 ( x ) = lim h 0 sin x cos h + cos x sin h - sin x h = lim h 0 sin x cos h - 1 h + cos x sin h h !
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