1. 5
Trigonometric Functions
Copyright © 2009 Pearson Addison-Wesley

2. Trigonometric Functions5
5.1 Angles
5.2 Trigonometric Functions
5.3 Evaluating Trigonometric Functions
5.4 Solving Right Triangles
Copyright © 2009 Pearson Addison-Wesley

3. Trigonometric Functions
5.2
Trigonometric Functions
Trigonometric Functions ▪ Quadrantal Angles ▪ Reciprocal Identities ▪ Signs and Ranges of Function Values ▪ Pythagorean Identities ▪ Quotient Identities
Copyright © 2009 Pearson Addison-Wesley 1.1-3

4. Trigonometric Functions
Let (x, y) be a point other the origin on the terminal side of an angle  in standard position. The distance from the point to the origin is
Copyright © 2009 Pearson Addison-Wesley

5. Trigonometric Functions
The six trigonometric functions of θ are defined as follows:
Copyright © 2009 Pearson Addison-Wesley 1.1-5

6. Example 1
FINDING FUNCTION VALUES OF AN ANGLE
The terminal side of angle  in standard position passes through the point (8, 15). Find the values of the six trigonometric functions of angle .
Copyright © 2009 Pearson Addison-Wesley 1.1-6

7. Example 1 FINDING FUNCTION VALUES OF AN ANGLE (continued)
Copyright © 2009 Pearson Addison-Wesley 1.1-7

8. Example 2
FINDING FUNCTION VALUES OF AN ANGLE
The terminal side of angle  in standard position passes through the point (–3, –4). Find the values of the six trigonometric functions of angle .
Copyright © 2009 Pearson Addison-Wesley 1.1-8

9. Use the definitions of the trigonometric functions.
Example 2
FINDING FUNCTION VALUES OF AN ANGLE (continued)
Use the definitions of the trigonometric functions.
Copyright © 2009 Pearson Addison-Wesley 1.1-9

10. Example 3
FINDING FUNCTION VALUES OF AN ANGLE
Find the six trigonometric function values of the angle θ in standard position, if the terminal side of θ is defined by x + 2y = 0, x ≥ 0.
We can use any point on the terminal side of  to find the trigonometric function values.
Choose x = 2.
Copyright © 2009 Pearson Addison-Wesley

11. Example 3
FINDING FUNCTION VALUES OF AN ANGLE (continued)
The point (2, –1) lies on the terminal side, and the corresponding value of r is
Multiply by to rationalize the denominators.
Copyright © 2009 Pearson Addison-Wesley

12. The terminal side passes through (0, 1). So x = 0, y = 1, and r = 1.
Example 4(a)
FINDING FUNCTION VALUES OF QUADRANTAL ANGLES
Find the values of the six trigonometric functions for an angle of 90°.
The terminal side passes through (0, 1). So x = 0, y = 1, and r = 1.
undefined
undefined
Copyright © 2009 Pearson Addison-Wesley

13. Example 4(b)
FINDING FUNCTION VALUES OF QUADRANTAL ANGLES
Find the values of the six trigonometric functions for an angle θ in standard position with terminal side through (–3, 0).
x = –3, y = 0, and r = 3.
undefined
undefined
Copyright © 2009 Pearson Addison-Wesley

14. Undefined Function Values
If the terminal side of a quadrantal angle lies along the y-axis, then the tangent and secant functions are undefined.
If the terminal side of a quadrantal angle lies along the x-axis, then the cotangent and cosecant functions are undefined.
Copyright © 2009 Pearson Addison-Wesley

15. Commonly Used Function Values
sin 
cos 
tan 
cot 
sec 
csc  0 1
undefined 1
undefined 1
undefined
90
undefined 1
180 1
undefined
270
undefined 1
360
Copyright © 2009 Pearson Addison-Wesley

16. Using a Calculator
A calculator in degree mode returns the correct values for sin 90° and cos 90°.
The second screen shows an ERROR message for tan 90° because 90° is not in the domain of the tangent function.
Copyright © 2009 Pearson Addison-Wesley

17. Caution
One of the most common errors involving calculators in trigonometry occurs when the calculator is set for radian measure, rather than degree measure.
Copyright © 2009 Pearson Addison-Wesley

18. Reciprocal Identities
For all angles θ for which both functions are defined,
Copyright © 2009 Pearson Addison-Wesley

19. Since cos θ is the reciprocal of sec θ,
Example 5(a)
USING THE RECIPROCAL IDENTITIES
Since cos θ is the reciprocal of sec θ,
Copyright © 2009 Pearson Addison-Wesley

20. Since sin θ is the reciprocal of csc θ,
Example 5(b)
USING THE RECIPROCAL IDENTITIES
Since sin θ is the reciprocal of csc θ,
Rationalize the denominator.
Copyright © 2009 Pearson Addison-Wesley

21. Signs of Function Values
 in Quadrant
sin 
cos 
tan 
cot 
sec 
csc  I + + + + + + +  II  +
III  + IV
Copyright © 2009 Pearson Addison-Wesley

22. Signs of Function Values
Copyright © 2009 Pearson Addison-Wesley

23. Example 6
IDENTIFYING THE QUADRANT OF AN ANGLE
Identify the quadrant (or quadrants) of any angle  that satisfies the given conditions.
(a) sin  > 0, tan  < 0.
Since sin  > 0 in quadrants I and II, and tan  < 0 in quadrants II and IV, both conditions are met only in quadrant II.
(b) cos  < 0, sec  < 0
The cosine and secant functions are both negative in quadrants II and III, so  could be in either of these two quadrants.
Copyright © 2009 Pearson Addison-Wesley

24. Ranges of Trigonometric Functions
Copyright © 2009 Pearson Addison-Wesley

25. Decide whether each statement is possible or impossible.
Example 7
DECIDING WHETHER A VALUE IS IN THE RANGE OF A TRIGONOMETRIC FUNCTION
Decide whether each statement is possible or impossible.
(a) sin θ = 2.5
Impossible
(b) tan θ =
Possible
(c) sec θ = .6
Impossible
Copyright © 2009 Pearson Addison-Wesley

26. Pythagorean Identities
For all angles θ for which the function values are defined,
Copyright © 2009 Pearson Addison-Wesley

27. Quotient Identities
For all angles θ for which the denominators are not zero,
Copyright © 2009 Pearson Addison-Wesley

28. Find sin θ and cos θ, given that and θ is in quadrant III.
Example 8
FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT
Find sin θ and cos θ, given that and θ is in quadrant III.
Since θ is in quadrant III, both sin θ and cos θ are negative.
Copyright © 2009 Pearson Addison-Wesley

29. Example 8
FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)
Caution
It is incorrect to say that sin θ = –4 and cos θ = –3, since both sin θ and cos θ must be in the interval [–1, 1].
Copyright © 2009 Pearson Addison-Wesley

30. Example 8
FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)
Use the identity to find sec θ. Then use the reciprocal identity to find cos θ.
Choose the negative square root since sec θ <0 for θ in quadrant III.
Secant and cosine are reciprocals.
Copyright © 2009 Pearson Addison-Wesley

31. Example 8
FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)
Choose the negative square root since sin θ <0 for θ in quadrant III.
Copyright © 2009 Pearson Addison-Wesley

32. Example 8
FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)
This example can also be worked by drawing θ in standard position in quadrant III, finding r to be 5, and then using the definitions of sin θ and cos θ in terms of x, y, and r.
Copyright © 2009 Pearson Addison-Wesley

33. Homework pp. 512–514 [1-22 , 53–60 , 63–112 (every third problem)]
Copyright © 2009 Pearson Addison-Wesley