Download presentation

Presentation is loading. Please wait.

Published byNyah Lawley Modified over 2 years ago

1
Derivative of the Sine Function By James Nickel, B.A., B.Th., B.Miss., M.A. www.biblicalchristianworldview.net

2
Basic Trigonometry Given the unit circle (radius = 1) with angle (theta), then, by definition, PA = sin and OA = cos . Copyright 2009 www.biblicalchristianworldview.net

3
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 www.biblicalchristianworldview.net

4
Question How does sin vary as varies? To find out, we must calculate the derivative of y = sin . Copyright 2009 www.biblicalchristianworldview.net

5
The Method of Increments We add to and determine what happens. We need to find the change in y or y. Copyright 2009 www.biblicalchristianworldview.net

6
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 www.biblicalchristianworldview.net

7
The Method of Increments Also remember that is just a “little bit” or infinitesimally small. Copyright 2009 www.biblicalchristianworldview.net

8
Our Goal Find QD = y To do this, we must show QDP ~ PAO We then let 0 Hence, Copyright 2009 www.biblicalchristianworldview.net

9
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 www.biblicalchristianworldview.net

10
Some Geometry Construct DP OA Note the transversal OP Hence, = DPO QPO = 90 DPO and DPQ are complementary. Hence, DPQ = 90 – Copyright 2009 www.biblicalchristianworldview.net

11
Some Geometry DPQ = 90 – DQP = Hence, QDP ~ PAO because both are right triangles and DQP = POA = Copyright 2009 www.biblicalchristianworldview.net

12
Some Geometry Consider DQ It represents the change in y (i.e., y) resulting from the change in (i.e., ). Copyright 2009 www.biblicalchristianworldview.net

13
Derivative Formula In QDP, cos = Note that as gets infinitesimally small, QP (the hypotenuse) converges to . Copyright 2009 www.biblicalchristianworldview.net

14
Derivative Formula Because QDP ~ PAO, then: Copyright 2009 www.biblicalchristianworldview.net

15
Derivative Formula Since PO = 1 and OA = cos , then: Copyright 2009 www.biblicalchristianworldview.net

16
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

17
Derivative Graph 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

18
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 www.biblicalchristianworldview.net

19
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 www.biblicalchristianworldview.net

20
Derivative Graph The cosine curve traces the derivative of the sine curve at every point on the sine curve. Copyright 2009 www.biblicalchristianworldview.net

Similar presentations

OK

Www.apcalculusbc.com. Consider the function We could make a graph of the slope: slope Now we connect the dots! The resulting curve is a cosine curve.

Www.apcalculusbc.com. Consider the function We could make a graph of the slope: slope Now we connect the dots! The resulting curve is a cosine curve.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on telephone exchange in indian railways Ppt on water scarcity solutions Ppt on limits and derivatives Ppt on different solid figures first grade Ppt on two point perspective lesson Ppt on non food crops Free ppt on human evolution Ppt on motivation for employees Ppt on computer viruses and worms Ppt on nuclear family and joint family band