1. Copyright © 2009 Pearson Addison-Wesley 1.3-1 1 Trigonometric Functions

2. Copyright © 2009 Pearson Addison-Wesley 1.3-2 1.1 Angles 1.2 Angle Relationships and Similar Triangles 1.3 Trigonometric Functions 1.4 Using the Definitions of the Trigonometric Functions 1 Trigonometric Functions

3. Copyright © 2009 Pearson Addison-Wesley1.1-3 1.3-3 Trigonometric Functions 1.3 Trigonometric Functions ▪ Quadrantal Angles

4. Copyright © 2009 Pearson Addison-Wesley 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

5. Copyright © 2009 Pearson Addison-Wesley1.1-5 1.3-5 Trigonometric Functions The six trigonometric functions of θ are defined as follows:

6. Copyright © 2009 Pearson Addison-Wesley1.1-6 1.3-6 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 FINDING FUNCTION VALUES OF AN ANGLE

7. Copyright © 2009 Pearson Addison-Wesley1.1-7 1.3-7 Example 1 FINDING FUNCTION VALUES OF AN ANGLE (continued)

8. Copyright © 2009 Pearson Addison-Wesley1.1-8 1.3-8 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 FINDING FUNCTION VALUES OF AN ANGLE

9. Copyright © 2009 Pearson Addison-Wesley1.1-9 1.3-9 Example 2 FINDING FUNCTION VALUES OF AN ANGLE (continued) Use the definitions of the trigonometric functions.

10. Copyright © 2009 Pearson Addison-Wesley1.1-10 1.3-10 Example 3 FINDING FUNCTION VALUES OF AN ANGLE We can use any point on the terminal side of  to find the trigonometric function values. Choose x = 2. 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.

11. Copyright © 2009 Pearson Addison-Wesley1.1-11 1.3-11 The point (2, –1) lies on the terminal side, and the corresponding value of r is Example 3 FINDING FUNCTION VALUES OF AN ANGLE (continued) Multiply by to rationalize the denominators.

12. Copyright © 2009 Pearson Addison-Wesley1.1-12 1.3-12 Example 4(a) FINDING FUNCTION VALUES OF QUADRANTAL ANGLES The terminal side passes through (0, 1). So x = 0, y = 1, and r = 1. undefined Find the values of the six trigonometric functions for an angle of 90°.

13. Copyright © 2009 Pearson Addison-Wesley1.1-13 1.3-13 Example 4(b) FINDING FUNCTION VALUES OF QUADRANTAL ANGLES x = –3, y = 0, and r = 3. undefined Find the values of the six trigonometric functions for an angle θ in standard position with terminal side through (–3, 0).

14. Copyright © 2009 Pearson Addison-Wesley1.1-14 1.3-14 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.

15. Copyright © 2009 Pearson Addison-Wesley 1.3-15 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 

16. Copyright © 2009 Pearson Addison-Wesley 1.3-16 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.

17. Copyright © 2009 Pearson Addison-Wesley1.1-17 1.3-17 Caution One of the most common errors involving calculators in trigonometry occurs when the calculator is set for radian measure, rather than degree measure.