math16B_suppl_notes_4

math16B_suppl_notes_4 - Math 16B S06 Supplementary Notes 4...

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Math 16B – S06 – Supplementary Notes 4 The Derivatives of sin t and cos t One can derive the formulas (1) d dt (sin t ) = cos t, d dt (cos t ) = - sin t starting from the relations (2) lim t 0 sin t t = 1 (3) lim t 0 cos t - 1 t = 0 . Note that (2) and (3) just say that (1) holds at the origin (since cos 0 = 1 and sin 0 = 0). Taking (2) and (3) temporarily for granted, let’s derive (1). By definition of the derivative, d dt (sin t ) = lim h 0 sin( t + h ) - sin t h . We use the addition formula sin( t + h ) = sin t cos h + cos t sin h to rewrite this as d dt (sin t ) = lim h 0 ± sin t ² cos h - 1 h ³ + cos t ² sin h h ³´ . By (2) and (3) the limit on the right side equals cos t , which establishes the first formula in (1). The second formula in (1) can be deduced from the first one by means of the identities cos t = sin µ t + π 2 , sin t = - cos µ t + π 2 and the chain rule. We have d dt (cos t ) = d dt µ sin µ t + π 2 ¶¶ = cos µ t + π 2 d dt µ t + π 2 = cos µ t + π 2 = - sin t. So, to establish (1), it only remains to establish (2) and (3). Once (2) is known (3) follows easily.
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math16B_suppl_notes_4 - Math 16B S06 Supplementary Notes 4...

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