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Trigonometry
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2. Copyright © Cengage Learning. All rights reserved.
6.4
GRAPHS OF SINE AND COSINE FUNCTIONS
Copyright © Cengage Learning. All rights reserved.

3. What You Should Learn
Sketch the graphs of basic sine and cosine functions.
Use amplitude and period to help sketch the graphs of sine and cosine functions.
Sketch translations of the graphs of sine and cosine functions.

4. Basic Sine and Cosine Curves

5. Basic Sine and Cosine Curves

6. Basic Sine and Cosine Curves
five key points to sketch the graphs:
the intercepts, maximum points, and minimum
Figure 6.49

7. Example 1 – Using Key Points to Sketch a Sine Curve
Sketch the graph of y = 2 sin x on the interval [–, 4].
Solution:
Intercept Maximum Intercept Minimum Intercept

8. Exercise
cont’d
Exercise 39

9. Amplitude and Period

10. Amplitude = value from x-axis to maximum/minimum
Amplitude and Period
Amplitude = value from x-axis to maximum/minimum

11. Example 2: Scaling: Vertical Shrinking and Stretching
Example 2 Exercise 41

12. Amplitude and Period
period: value to complete 1 cycle, from 1st intercept to3rd intercept.

13. Example 3 – Scaling: Horizontal Stretching
Sketch the graph of
Solution:
The amplitude is 1. Moreover, because b = , the period is
Substitute for b.

14. Example 3 – Solution
cont’d
divide the period-interval [0, 4] into four equal parts with the values , 2, and 3 to obtain the key points on the graph.
Intercept Maximum Intercept Minimum Intercept
(0, 0), (, 1), (2, 0), (3, –1), and (4, 0)
The graph is shown in Figure 6.53.
Figure 6.53

15. Translations of Sine and Cosine Curves

16. Translations of Sine and Cosine Curves

17. Example 5 – Horizontal Translation
Sketch the graph of
y = –3 cos(2 x + 4).
Solution:
The amplitude is 3 and the period is 2 / 2 = 1.
By solving the equations
2 x + 4 = 0
2 x = –4
x = –2
and
2 x + 4 = 2

18. Example 5 – Solution 2 x = –2 x = –1
cont’d
2 x = –2
x = –1
you see that the interval [–2, –1] corresponds to one cycle of the graph.
Dividing this interval into four equal parts produces the key points
Minimum Intercept Maximum Intercept Minimum
and

19. Example 5 – Solution The graph is shown in Figure 6.55. cont’d

20. Translations of Sine and Cosine Curves
vertical translation for :
y = d + a sin(bx – c)
and
y = d + a cos(bx – c).
Shift d units upward for d > 0
Shift d units downward for d < 0
EXP 6

21. Graph of the Tangent Function

22. Graph of the Tangent Function
Figure 6.59

23. Introduction
We will learn how to use the fundamental identities to do the following.
1. Evaluate trigonometric functions.
2. Simplify trigonometric expressions.

24. Introduction

25. Introduction
cont’d

26. Example 2 – Simplifying a Trigonometric Expression
Simplify sin x cos2 x – sin x.
Solution:
First factor out a common monomial factor and then use a fundamental identity.
sin x cos2 x – sin x = sin x (cos2 x – 1)
= –sin x(1 – cos2 x)
= –sin x(sin2 x)
= –sin3 x
Factor out common monomial factor.
Factor out –1.
Pythagorean identity.
Multiply.

27. Example 3 – Factoring Trigonometric Expressions

28. Example 4 – Factoring Trigonometric Expressions

29. Example 5 – Simplifying Trigonometric Expressions

30. Example 6 – Adding Trigonometric Expressions

31. Example 7 – Rewriting a Trigonometric Expression
Rewrite so that it is not in fractional form.
Solution:
From the Pythagorean identity cos2 x = 1 – sin2 x = (1 – sin x)(1 + sin x), you can see that multiplying both the numerator and the denominator by (1 – sin x) will produce a monomial denominator.
Multiply numerator and
denominator by (1 – sin x).
Multiply.

32. Example 7 – Solution cont’d Pythagorean identity.
Write as separate fractions.
Product of fractions.
Reciprocal and quotient identities.

33. Example 8 – Trigonometric Substitution

34. Example 9 – Rewriting a Logarithmic Expression

35. What You Should Learn
Verify trigonometric identities.

36. Verifying Trigonometric Identities

37. Example 1 – Verifying a Trigonometric Identity
Verify the identity (sec2  – 1) / (sec2  ) = sin2 .
Solution:
Pythagorean identity
Simplify.
Reciprocal identity
Quotient identity
Simplify.

38. Example 1 – Solution
cont’d
Notice how the identity is verified. You start with the left side of the equation (the more complicated side) and use the fundamental trigonometric identities to simplify it until you obtain the right side.
Simplify.

39. Example 2 – Verifying a Trigonometric Identity

40. Example 3 – Verifying a Trigonometric Identity

41. Example 4 – Converting to Sines and Cosines

42. Example 5 – Verifying a Trigonometric Identity

43. Example 6 – Verifying a Trigonometric Identity

44. Example 7 – Verifying a Trigonometric Identity

45. Multiple-Angle Formulas

46. What You Should Learn
Use multiple-angle formulas to rewrite and evaluate trigonometric functions.
Use half-angle formulas to rewrite and evaluate trigonometric functions.

47. Multiple-Angle Formulas

48. Example 1 – Solving a Multiple-Angle Equation
Solve 2 cos x + sin 2x = 0.
Solution: Begin by rewriting the equation so that it involves functions of x (rather than 2x). Then factor and solve.
2 cos x + sin 2x = 0
2 cos x + 2 sin x cos x = 0
2 cos x(1 + sin x) = 0
2 cos x = 0 and 1 + sin x = 0
Write original equation.
Double-angle formula.
Factor.
Set factors equal to zero.

49. Example 1 – Solution So, the general solution is and
cont’d
So, the general solution is
and
where n is an integer. Try verifying these solutions graphically.
Solutions in [0, 2)

50. Half-Angle Formulas

51. Half-Angle Formulas

52. Example 6 – Using a Half-Angle Formula
Find the exact value of sin 105.
Solution: Begin by noting that 105 is half of 210. Then, using the half-angle formula for sin(u / 2) and the fact that 105 lies in Quadrant II, you have

53. Example 6 – Solution
cont’d .
The positive square root is chosen because sin  is positive in Quadrant II.

54. Product-to-Sum Formulas

55. Product-to-Sum Formulas

56. Example 8 – Writing Products as Sums
Rewrite the product cos 5x sin 4x as a sum or difference.
Solution:
Using the appropriate product-to-sum formula, you obtain
cos 5x sin 4x = [sin(5x + 4x) – sin(5x – 4x)]
= sin 9x – sin x.