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Derivative of the Sine Function By James Nickel, B.A., B.Th., B.Miss., M.A. www.biblicalchristianworldview.net Basic Trigonometry • Given the unit circle (radius = 1) with angle (theta), then, by definition, PA = sin and OA = cos . P 1 sin O cos Copyright 2009 www.biblicalchristianworldview.net A 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. P 1 sin O cos Copyright 2009 www.biblicalchristianworldview.net A Question • How does sin vary as varies? • To find out, we must calculate the derivative of y = sin. Copyright 2009 www.biblicalchristianworldview.net The Method of Increments • We add to and determine what happens. • We need to find the change in y or y. y 1 P sin O cos Copyright 2009 www.biblicalchristianworldview.net A 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. Q P 1 1 O Copyright 2009 www.biblicalchristianworldview.net B A The Method of Increments • Also remember that is just a “little bit” or infinitesimally small. Q P 1 1 O Copyright 2009 www.biblicalchristianworldview.net B A Our Goal • Find QD = y • To do this, we must show QDP ~ PAO • We then let 0 • Hence, y DQ OA PO Q P D 1 1 O Copyright 2009 www.biblicalchristianworldview.net B A 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 Q P D 1 1 l O Copyright 2009 www.biblicalchristianworldview.net B A Some Geometry Construct DP OA Note the transversal OP Hence, = DPO QPO = 90 DPO and DPQ are complementary. • Hence, DPQ = 90 – • • • • Q P D 1 1 sin O Copyright 2009 www.biblicalchristianworldview.net B A l Some Geometry • DPQ = 90 – DQP = • Hence, QDP ~ PAO because both are right triangles and DQP = POA = Q P D 1 1 l O Copyright 2009 www.biblicalchristianworldview.net B A Some Geometry • Consider DQ • It represents the change in y (i.e., y) resulting from the change in (i.e., ). y DQ sin sin Q P D 1 1 sin O Copyright 2009 www.biblicalchristianworldview.net B A l Derivative Formula y DQ sin sin DQ In QDP, cos = Note that as gets infinitesimally small, QP (the hypotenuse) converges to . Q P D 1 1 sin O Copyright 2009 www.biblicalchristianworldview.net B A l Derivative Formula • Because QDP ~ PAO, then: DQ OA PO Q P D 1 1 sin O Copyright 2009 www.biblicalchristianworldview.net B A l Derivative Formula • Since PO = 1 and OA = cos , then: y DQ sin sin OA cos cos PO 1 Q P D 1 1 sin O B OA=cos Copyright 2009 www.biblicalchristianworldview.net A l 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 www.biblicalchristianworldview.net Derivative Graph y-axis -axis y = sin • The solid line graph represents y = sin (the sine curve). • The dotted line graph represents y = cos (the cosine curve). Copyright 2009 www.biblicalchristianworldview.net Derivative Graph y-axis y = sin -axis • 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 www.biblicalchristianworldview.net Derivative Graph y-axis -axis y = sin • 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 www.biblicalchristianworldview.net Derivative Graph y-axis -axis y = sin • The cosine curve traces the derivative of the sine curve at every point on the sine curve. Copyright 2009 www.biblicalchristianworldview.net