1. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-1 Graphs of the Sine and Cosine Functions 4.1 Periodic Functions ▪ Graph of the Sine Function▪ Graph of the Cosine Function ▪ Graphing Techniques, Amplitude, and Period

2. Formulas to Memorize for Chapter 4 y = a sin b(x + c) + dy = a cos b(x + c) + d y = a tan b(x + c) + d y = a csc b(x + c) + dy = a sec b(x + c) + d y = a cot b(x + c) + d SOH CAH TOA sin starts @ zerocos starts @ the a value zero, asy, zero, asy,… Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-2 AmplitudePeriodTable of values Sine Cosine │a│ 2π/b0 < b(x + c) < 2π Cut in ¼’s Cosecant Secant 2π/b0 < b(x + c) < 2π Cut in ¼’s Tangent Cotangent π/b0 < b(x + c) < π Cut in ¼’s

3. The Unit Circle Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-3 Cos 0 = Cos π / 2 = Cos 3π / 2 = Cos π = Cos 2π = Cos 5π / 2 = Cos 7π / 2 = Cos 3π = Cos 4π = Cos 9π / 2 = 1 0 0 1 0 0 1 0

4. The Unit Circle Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-4 Sin 0 = Sin π / 2 = Sin 3π / 2 = Sin π = Sin 2π = Sin 5π / 2 = Sin 7π / 2 = Sin 3π = Sin 4π = Sin 9π / 2 = 0 1 0 0 1 0 0 1

5. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-5 and compare to the graph of y = sin x. 4.1 Example 1 Graphing y = a sin x (page 147) The shape of the graph is the same as the shape of y = sin x except it is ½ as steep. The range of is x0π/2π3π/22π sin x0100 ½ sin x0½0-½-½0

6. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. and compare to the graph of y = sin x. 4.1 Example 2 Graphing y = sin bx (page 148) The coefficient of x is, so b =, and the period is Divide the interval into four equal parts to get the x-values that will yield minimum and maximum points and x-intercepts. Period = x03π/23π9π/26π sin 1/3 x0100

7. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-7 4.1 Example 3 Graphing y = cos bx (page 149) The coefficient of x is, so b =, and the period is Divide the interval into four equal parts to get the x-values that will yield minimum and maximum points and x-intercepts. x0π2π3π4π cos 1/2 x1001 Period =

8. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-8 Graph y = –3 sin 2x. 4.1 Example 4 Graphing y = a sin bx (page 150) The coefficient of x is 2, so b = 2, and the period is The amplitude is |–3| = 3. Divide the interval into four equal parts to get the x-values that will yield minimum and maximum points and x-intercepts. x0π/4π/23π/4π sin 2x0100 -3 sin 2x0-3030 Period =Amplitude = |a|

9. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-9 4.1 Example 5 Graphing y = a cos bx for b Equal to a Multiple of π (page 151) The amplitude is |2| = 2. Divide the interval [0, 4] into four equal parts to get the x-values that will yield minimum and maximum points and x-intercepts. The coefficient of x is, so b =, and the period is x01234 cos π/2 x1001 2cos π/2 x20-202 Period =Amplitude = |a|

10. Summary: Formulas to Memorize for Chapter 4 y = a sin b(x + c) + dy = a cos b(x + c) + d y = a tan b(x + c) + d y = a csc b(x + c) + dy = a sec b(x + c) + d y = a cot b(x + c) + d SOH CAH TOA sin starts @ zerocos starts @ the a value zero, asy, zero, asy,… Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-10 AmplitudePeriodTable of values Sine Cosine │a│ 2π/b0 < b(x + c) < 2π Cut in ¼’s Cosecant Secant 2π/b0 < b(x + c) < 2π Cut in ¼’s Tangent Cotangent π/b0 < b(x + c) < π Cut in ¼’s 0,1,0,-1,01,0,-1,0,1

11. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-11 Translations of the Graphs of the Sine and Cosine Functions 4.2 Horizontal Translations ▪ Vertical Translations ▪ Combinations of Translations

12. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-12 4.2 Example 1 Graphing y = sin ( x – d ) (page 158) Amplitude = 1 Phase shift : left 3π / 4 = - 3π / 4 Vertical translation: none Period = 2π x-3π/4-π/4π/43π/45π/4 sin(x + 3π/4)0100

13. 4-13 4.2 Example 2 Graphing y = a cos ( x – d ) (page 159) Amplitude = 2Phase shift : right π / 3 = π / 3 Vertical translation: nonePeriod = 2π xπ/35π/64π/311π/67π/3 cos(x - π/3)1001 -2cos(x - π/3)-2020

14. 4.2 Example 3 Graphing y = a cos b ( x – d ) (page 160) Write the equation in the form y = a cos b(x – c) Amplitude = 3 / 2 Phase shift : right π / 2 = π / 2 Vertical translation: none Period = π xπ/23π/4π5π/43π/2 cos2(x - π/2)1001 3 / 2 cos(x - π/3)3/20-3/203/2 y = 3 / 2 cos 2(x – π / 2 )

15. 4-15 4.2 Example 4 Graphing y = c + a cos bx (page 161) Graph y = –2 + 3 cos 2x over two periods. Amplitude = 3 Phase shift : none Vertical translation: 2 units down = -2 Period = π Re-write equation: y = 3 cos (2x) – 2 x0π/4π/23π/4π cos 2x1001 3cos 2x30-303 3cos 2x – 21-2-5-21

16. 4.2 Example 5 Graphing y = c + a sin b ( x – d ) (page 162) xπ/3π/22π/35π/6π sin 3(x - π/3)0100 -2sin 3(x - π/3)0-2020 -2sin3(x - π/3) + 442464 Amplitude = 2 Period = 2π / 3 Phase shift : π / 3 to the right = π / 3 Vertical translation: 4 units up = 4 Re-write equation: y = -2 sin 3(x – π / 3 ) + 4 0 < 3x – π < 2π

17. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-17 Translations of the Graphs of the Sine and Cosine Functions 4.2 Horizontal Translations Vertical Translations Combinations of Translations Reminder for Sine and Cosine graphs: period = 2π / b amplitude = |a| sin … 0,1,0,-1,0 cos… 1,0,-1,0,1

18. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-18 Graphs of the Tangent and Cotangent Functions 4.3 Graph of the Tangent Function ▪ Graph of the Cotangent Function ▪ Graphing Techniques

19. The Unit Circle Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-19 Tan 0 = Tan π / 2 = Tan 3π / 2 = Tan π = Tan 2π = Tan 3π / 4 = Tan π / 4 = Tan 5π / 4 = Tan 7π / 4 = 0 undefined 0 0 1 1

20. x03π/83π/49π/83π/2 Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-20 4.3 Example 1 Graphing y = tan bx (page 171) Period = 0 < 2 / 3 x < π 0 < x < 3 / 2 π x03π/83π/49π/83π/2 tan ( 2 / 3 x)0Asy0 x-15π/8-3π/2-9π/8-3π/4-3π/8 tan ( 2 / 3 x)01Asy0 tan ( 2 / 3 x)01Asy 1

21. 4.3 Example 2 Graphing y = a tan bx (page 172) 0 < 2x < π 0 < x < π / 2 x0π/8π/43π/8π/2 tan (2x)0Asy0 -1 / 2 tan (2x)0-½Asy½0 x0π/8π/43π/8π/2 -1 / 2 tan (2x)0-½Asy½0 x-5π/8-π/2-3π/8-π/4-π/8 -1 / 2 tan (2x)½0-½Asy½ 1

22. 4-22 4.3 Example 3 Graphing y = a cot bx (page 172) 0 < ½ x < π 0 < x < 2π x0π/2π3π/22π cot (1/2 x)Asy0 3cot (1/2 x)Asy30-3Asy The pattern to repeat 1

23. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-23 4.3 Example 4 Graphing a Tangent Function With a Vertical Translation (page 172) 0 < x < π x0π/4π/23π/4π tan x0Asy0 tan (x) – 3-3-2Asy-4-3 The pattern to repeat 1 π Re-write equation: y = tan (x) – 3 1

24. 4-24 4.3 Example 5 Graphing a Cotangent Function With Vertical and Horizontal Translations (page 173) Re-write equation: y = cot (x + π/2) + 3 x -π/2-π/2 -π/4-π/4 0 π/4π/4 π/2π/2 cot (x + π / 2 ) Asy0 cot (x + π / 2 ) + 3 Asy432 The pattern to repeat 0 < x + π / 2 < π -π / 2 < x < π / 2 1

25. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-25 Graphs of the Tangent and Cotangent Functions 4.3 Graph of the Tangent Function Graph of the Cotangent Function Graphing Techniques Reminder for Tangent and Cotangent graphs: period = π / b amplitude = none tangent … 0, 1, asy, -1, 0 cotangent … asy, 1, 0, -1, asy

26. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-26 Graphs of the Secant and Cosecant Functions 4.4 Graph of the Secant Function ▪ Graph of the Cosecant Function ▪ Graphing Techniques ▪ Addition of Ordinates ▪ Connecting Graphs with Equations

27. 4-27 4.4 Example 1 Graphing y = a sec bx (page 180) Graph the corresponding reciprocal function y = 3 cos 2x. Graph y = 3 sec 2x. The vertical asymptotes of y = 3 sec 2x are at the x-intercepts of y = 3 cos 2x Continue the pattern to the left and there are also v.asy @ x = -π / 4 and x = -3π / 4 x0π/4π/23π/4π cos 2x1001 sec 2x1AsyAsy1 3 sec 2x3Asy-3Asy3 3cos 2x30-303

28. 4-28 4.4 Example 2 Graphing y = a csc ( x – d ) (page 181) Graph the corresponding reciprocal function The vertical asymptotes of y = ½ csc(x + π / 4 ) are at the x-intercepts of y = ½ sin(x + π / 4 ) x = -π / 4, x = 3π / 4, and x = 7π / 4 Continuing this pattern to the left, there are also v. asymptotes at x = -5π / 4 and x = -7π / 4 x-π/45π/83π/213π/87π/4 sin (x + π/4)0100 csc(x + π/4)Asy1 Asy ½ csc(x + π/4)Asy½ -½Asy ½ sin (x + π/4)0½0-½0

29. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-29 4.4 Example 3(a) Determining an Equation for a Graph (page 182) Determine an equation for the graph. This is the graph of y = tan x, stretched vertically by a factor of 3. y = 3 tan x

30. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-30 4.4 Example 3(b) Determining an Equation for a Graph (page 182) Determine an equation for the graph. This is the graph of y = cot x, but the period is instead of

31. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-31 4.4 Example 3(c) Determining an Equation for a Graph (page 182) Determine an equation for the graph. This is the graph of y = csc x translated one unit down. y = –1 + csc x

32. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-32 Harmonic Motion 4.5 Simple Harmonic Motion ▪ Damped Oscillatory Motion

33. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-33 4.5 Example 1 Modeling the Motion of a Spring (page 186) Suppose that an object is attached to a coiled spring. It is pulled down a distance of 16 cm from its equilibrium position and then released. The time for one complete oscillation is 6 seconds. (a)Give an equation that models the position of the object at time t. Starting position at t = 0 is -16 cm → a = -16 Time to complete one oscillation is 6 sec 6 = 2π / b → b = π / 3

34. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-34 4.5 Example 1(b, c) Modeling the Motion of a Spring (page 186) (b)Determine the position at t = 1.5 seconds. At t = 1.5 seconds, the object is at the equilibrium position. (c)Find the frequency. The frequency is the reciprocal of the period.

35. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 4-35 4.5 Example 2 Analyzing Harmonic Motion (page 187) Suppose that an object oscillates according to the model s(t) = 2.5 sin 5t, where t is in seconds and s(t) is in meters. Analyze the motion. a = 2.5, so the object oscillates 2.5 meters in either direction from the starting point. and The motion is harmonic because the model is of the form