1. 5.2-1 5.2Trigonometric Functions Let (x, y) be a point other then the origin on the terminal side of an angle  in standard position. The distance from the point to the origin is

2. 5.2-2 The six trigonometric functions of θ are defined as follows:

3. 5.2-3 The terminal side of angle  in standard position passes through the point (8, 15). Find the values of the six trigonometric functions of angle . Example 1

4. 5.2-4 The terminal side of angle  in standard position passes through the point (–3, –4). Find the values of the six trigonometric functions of angle . Example 2

5. 5.2-5 Example 3 We can use any point on the terminal side of  to find the trigonometric function values. 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.

6. 5.2-6 Example 4 If the terminal side of angle θ in standard position lies on the line defined by 3x + 4y = 0, x ≥ 0. Then the value of 5sin  + 10cos  is A)5 B)9.5 C)11 D)-4 E)-3

7. 5.2-7 Example 5(a) FINDING FUNCTION VALUES OF QUADRANTAL ANGLES Find the values of the six trigonometric functions for an angle of 90°.

8. 5.2-8 Example 5(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).

9. 5.2-9 Commonly Used Function Values undefined1 010 360  11 undefined0 0 11 270  undefined 11 0 11 0 180  1undefined0 01 90  undefined1 010 00 csc  sec  cot  tan  cos  sin 

10. 5.2-10 Reciprocal Identities For all angles θ for which both functions are defined,

11. 5.2-11 Example 5 USING THE RECIPROCAL IDENTITIES

12. 5.2-12 Signs of Function Values  +  +  IV  ++  III +  +II ++++++I csc  sec  cot  tan  cos  sin   in Quadrant

13. 5.2-13 Signs of Function Values

14. 5.2-14 Identify the quadrant (or quadrants) of any angle  that satisfies the given conditions. Example 6 IDENTIFYING THE QUADRANT OF AN ANGLE (a) sin  > 0, tan  < 0. (b) cos  < 0, sec  < 0

15. 5.2-15 Ranges of Trigonometric Functions

16. 5.2-16 Decide whether each statement is possible or impossible. Example 7 DECIDING WHETHER A VALUE IS IN THE RANGE OF A TRIGONOMETRIC FUNCTION (a) sin θ = 2.5 (b) tan θ = 110.47 (c) sec θ = 0.6

17. 5.2-17 Pythagorean Identities For all angles θ for which the function values are defined,

18. 5.2-18 Quotient Identities For all angles θ for which the denominators are not zero,

19. 5.2-19 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.

20. 5.2-20 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]. Example 8 FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)

21. 5.2-21 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.