1. Copyright © 2007 Pearson Education, Inc. Slide 8-1 5.4Evaluating Trigonometric Functions Acute angle A is drawn in standard position as shown. Right-Triangle-Based Definitions of Trigonometric Functions For any acute angle A in standard position,

2. Copyright © 2007 Pearson Education, Inc. Slide 8-2 5.4 Finding Trigonometric Function Values of an Acute Angle in a Right Triangle Example Find the values of sin A, cos A, and tan A in the right triangle. Solution –length of side opposite angle A is 7 –length of side adjacent angle A is 24 –length of hypotenuse is 25

3. Copyright © 2007 Pearson Education, Inc. Slide 8-3 Angles that deserve special study are 30 º, 45 º, and 60 º. 5.4 Trigonometric Function Values of Special Angles Using the figures above, we have the exact values of the special angles summarized in the table on the right.

4. Copyright © 2007 Pearson Education, Inc. Slide 8-4 In a right triangle ABC, with right angle C, the acute angles A and B are complementary. Since angles A and B are complementary, and sin A = cos B, the functions sine and cosine are called cofunctions. Similarly for secant and cosecant, and tangent and cotangent. 5.4 Cofunction Identities

5. Copyright © 2007 Pearson Education, Inc. Slide 8-5 NoteThese identities actually apply to all angles (not just acute angles). 5.4 Cofunction Identities If A is an acute angle measured in degrees, then If A is an acute angle measured in radians, then

6. Copyright © 2007 Pearson Education, Inc. Slide 8-6 5.4 Reference Angles A reference angle for an angle , written , is the positive acute angle made by the terminal side of angle  and the x-axis. ExampleFind the reference angle for each angle. (a)218 º (b) Solution (a)  = 218 º – 180 º = 38 º (b)

7. Copyright © 2007 Pearson Education, Inc. Slide 8-7 5.4 Special Angles as Reference Angles ExampleFind the values of the trigonometric functions for 210 º. SolutionThe reference angle for 210 º is 210 º – 180 º = 30 º. Choose point P on the terminal side so that the distance from the origin to P is 2. A 30 º - 60 º right triangle is formed.

8. Copyright © 2007 Pearson Education, Inc. Slide 8-8 5.4 Finding Trigonometric Function Values Using Reference Angles ExampleFind the exact value of each expression. (a)cos(–240 º )(b) tan 675 º Solution (a)–240 º is coterminal with 120 º. The reference angle is 180 º – 120 º = 60 º. Since –240 º lies in quadrant II, the cos(–240 º ) is negative. (b) Similarly, tan 675 º = tan 315 º = –tan 45 º = –1.

9. Copyright © 2007 Pearson Education, Inc. Slide 8-9 5.4 Finding Trigonometric Function Values with a Calculator ExampleApproximate the value of each expression. (a)cos 49 º 12(b) csc 197.977 º SolutionSet the calculator in degree mode.

10. Copyright © 2007 Pearson Education, Inc. Slide 8-10 5.4 Finding Angle Measure ExampleUsing Inverse Trigonometric Functions to Find Angles (a)Use a calculator to find an angle  in degrees that satisfies sin  .9677091705. (b)Use a calculator to find an angle  in radians that satisfies tan  .25. Solution (a)With the calculator in degree mode, we find that an angle having a sine value of.9677091705 is 75.4º. Write this as sin -1.9677091705  75.4º. (b)With the calculator in radian mode, we find tan -1.25 .2449786631.

11. Copyright © 2007 Pearson Education, Inc. Slide 8-11 5.4 Finding Angle Measure ExampleFind all values of , if  is in the interval [0º, 360º) and SolutionSince cosine is negative,  must lie in either quadrant II or III. Since So the reference angle  = 45º. The quadrant II angle  = 180º – 45º = 135º, and the quadrant III angle  = 180º + 45º = 225º.