1. Copyright © Cengage Learning. All rights reserved.6
Trigonometry
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2. Copyright © Cengage Learning. All rights reserved.
6.1
ANGLES AND THEIR MEASURE
Copyright © Cengage Learning. All rights reserved.

3. What You Should Learn Use degree measure. Use radian measure.
Convert between degree and radian measures.

4. Degree Measure

5. Degree Measure Quadrants: Type of angle: Right angle:
quarter revolution
Acute angle: between 0 and 90
Obtuse angle:
between 90 and 180
Full revolution
Straight angle: half revolution

6. Degree Measure
Two positive angles  and  are complementary (complements of each other) if their sum is 90.
Two positive angles are supplementary (supplements of each other) if their sum is 180. See Figure 6.11.
Complementary angles
Supplementary angles

7. Radian Measure

8. Arc length,s = radius, r when  = 1 radian
Radian Measure
Arc length,s = radius, r when  = 1 radian

9. Conversion of Angle Measure

10. Conversion of Angle Measure
From the latter equation, you obtain
and
Which lead to the conversion rules.

11. Conversion of Angle Measure
When no units of angle measure are specified, radian measure is implied.
For instance, if you write  = 2, you imply that  = 2 radians.
Figure 6.17

12. Example 4 – Converting from Degrees to Radiansb. c.
Multiply by  / 180.
Multiply by  / 180.
Multiply by  / 180.

13. Exercise
Exercise 63

14. Example 5: Converting from Radians to Degrees
Page 447 Exercise 67

15. Applications

16. Applications
The radian measure formula,  = s / r, can be used to measure arc length along a circle.

17. Example 6
Page 449 Exercise 93

18. Applications

19. Example 9 – Area of a Sector of a Circle
A sprinkler on a golf course fairway sprays water over a distance of 70 feet and rotates through an angle of 120 (see Figure 6.22). Find the area of the fairway watered by the sprinkler.
Figure 6.22

20. Example 9 – Solution First convert 120 to radian measure as follows.
 = 120
Then, using  = 2 /3 and r = 70, the area is
Multiply by  /180.
Formula for the area of a
sector of a circle.

21. Example 9 – Solution cont’d Substitute for r and . Simplify.

22. Exercise
Exercise 115

23. What You Should Learn
Evaluate trigonometric functions of acute angles.
Use fundamental trigonometric identities.
Use a calculator to evaluate trigonometric functions.

24. The Six Trigonometric Functions

25. The Six Trigonometric Functions
From a right triangle : (page 456)

26. Example 1 – Evaluating Trigonometric Functions
Use the triangle in Figure 6.24 to find the values of the six trigonometric functions of .
Solution:
By the Pythagorean Theorem,
(hyp)2 = (opp)2 + (adj)2, it follows that
Figure 6.24

27. Example 1 – Solution So, the six trigonometric functions of  are
cont’d
So, the six trigonometric functions of  are

28. Exercise
Use the Pythagorean Theorem to find the third side of the triangle. Hence, find the values of the six trigonometric functions of the angle 1. 3. 2. 4.

29. Example 2 – Evaluating Trigonometric Functions of 45˚
Figure 6.24

30. Example 3 – Evaluating Trigonometric Functions of 30˚ and 60˚.

31. The Six Trigonometric Functions

32. The Six Trigonometric Functions
If  is an acute angle, the following relationships are true.
sin(90 –  ) = cos  cos(90 –  ) = sin 
tan(90 –  ) = cot  cot(90 –  ) = tan 
sec(90 –  ) = csc  csc(90 –  ) = sec 

33. Trigonometric Identities

34. Trigonometric Identities
Note: sin2 =(sin)2
cos2 =(cos  )2

35. Example 4 – Applying Trigonometric Identities
Let  be an acute angle such that sin  = 0.6. Find the values of
cos 
tan 
using trigonometric identities.
Solution:
a. use the Pythagorean identity
sin2  + cos2  = 1.
(0.6)2 + cos2  = 1
cos2  = 1 – (0.6)2
= 0.64
Substitute 0.6 for sin .
Subtract (0.6)2 from each side.

36. Example 4 – Solution cos  = = 0.8. b. = 0.75. cont’d
Extract the positive square root.
(Given in question  an acute angle)
= 0.75.

37. Exercise

38. Example 5: Applying Trigonometric Identities

39. Exercise

40. Evaluating Trigonometric Functions with a Calculator

41. Evaluating Trigonometric Functions with a Calculator
To evaluate csc( /8), use the fact that
and enter the following keystroke sequence in radian mode.
Display

42. Example 6 – Using a Calculator
Function Mode Calculator Keystrokes Display
a. sin 76.4 Degree
b. cot Radian or
Display

43. What You Should Learn Find reference angles.
Evaluate trigonometric functions of real numbers.

44. Reference Angles

45. Reference Angles
The values of the trigonometric functions of angles greater than 90 (or less than 0) can be determined from their values at corresponding acute angles called reference angles. .
 ′ =  –   ′ = 180 – 
 ′ =  –   ′ =  – 180
 ′ = 2 –   ′ = 360 – 

46. Example 4 – Finding Reference Angles
Find the reference angle  ′.
a.  = 300
b.  = 2.3
c.  = –135

47. Example 4(a) – Solution a)  ′ = 360 – 300 = 60 b)  ′ =  – 2.3
a)  ′ = 360 – 300
= 60

48. Example 4(c) – Solution
cont’d
 ′ = 225 – 180
= 45.

49. Exercise