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Derivative of the Sine Function By James Nickel, B.A., B.Th., B.Miss., M.A.

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Basic Trigonometry Given the unit circle (radius = 1) with angle (theta), then, by definition, PA = sin and OA = cos . Copyright 2009

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Basic Trigonometry As changes from 0 to /2 radians (90 ), sin also changes from 0 to 1. As changes from 0 to /2 radians, cos also changes from 1 to 0. Copyright 2009

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Question How does sin vary as varies? To find out, we must calculate the derivative of y = sin . Copyright 2009

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The Method of Increments We add to and determine what happens. We need to find the change in y or y. Copyright 2009

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The Method of Increments PA has increased to QB where QB = sin( + ). OA has decreased to OB where OB = cos( + ). also represents the radian measure of the arc from P to Q. Copyright 2009

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The Method of Increments Also remember that is just a “little bit” or infinitesimally small. Copyright 2009

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Our Goal Find QD = y To do this, we must show QDP ~ PAO We then let 0 Hence, Copyright 2009

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Some Geometry Let’s now draw the tangent line l to the circle at point P. We know that l PO (the tangent line intersects the circle’s radius at a right angle). Hence, QPO = 90 Copyright 2009

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Some Geometry Construct DP OA Note the transversal OP Hence, = DPO QPO = 90 DPO and DPQ are complementary. Hence, DPQ = 90 – Copyright 2009

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Some Geometry DPQ = 90 – DQP = Hence, QDP ~ PAO because both are right triangles and DQP = POA = Copyright 2009

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Some Geometry Consider DQ It represents the change in y (i.e., y) resulting from the change in (i.e., ). Copyright 2009

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Derivative Formula In QDP, cos = Note that as gets infinitesimally small, QP (the hypotenuse) converges to . Copyright 2009

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Derivative Formula Because QDP ~ PAO, then: Copyright 2009

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Derivative Formula Since PO = 1 and OA = cos , then: Copyright 2009

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Derivative Formula As we let approach 0 as a limit, then: y (sin ) = cos This means as increases, sin increases at a instantaneous rate of cos . Copyright 2009

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Derivative Graph The solid line graph represents y = sin (the sine curve). The dotted line graph represents y = cos (the cosine curve). Copyright 2009

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Derivative Graph Note that when = 0, then cos = 1. This means that the slope of the line tangent to the sine curve at 0 is 1. The cosine curve plots that derivative. Copyright 2009

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Derivative Graph When = /2, then cos = 0. This means that the slope of the line tangent to the sine curve at /2 is 0 (the slope is parallel to the -axis). Copyright 2009

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Derivative Graph The cosine curve traces the derivative of the sine curve at every point on the sine curve. Copyright 2009

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