Chapter 8: Trigonometric Functions And Applications

2. Chapter 8: Trigonometric Functions And Applications 8.1 Angles and Their Measures 8.2 Trigonometric Functions and Fundamental Identities 8.3 Evaluating Trigonometric Functions 8.4 Applications of Right Triangles 8.5 The Circular Functions 8.6 Graphs of the Sine and Cosine Functions 8.7 Graphs of the Other Circular Functions 8.8 Harmonic Motion

3. 8.3 Evaluating 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,

4. 8.3 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

5. 8.3 Trigonometric Function Values of Special Angles • Angles that deserve special study are 30º, 45º, and 60º. Using the figures above, we have the exact values of the special angles summarized in the table on the right.

6. 8.3 Cofunction Identities • 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.

7. 8.3 Cofunction Identities If A is an acute angle measured in degrees, then If A is an acute angle measured in radians, then Note These identities actually apply to all angles (not just acute angles).

8. 8.3 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. Example Find the reference angle for each angle. • 218º (b) Solution (a)   = 218º – 180º = 38º (b)

9. 8.3 Special Angles as Reference Angles Example Find the values of the trigonometric functions for 210º. Solution The reference angle for 210º is 210º – 180º = 30º. Choose point P on the terminal side so that the distance from the origin to P is 2. A 30º - 60º right triangle is formed.

10. 8.3 Finding Trigonometric Function Values Using Reference Angles Example Find the exact value of each expression. • cos(–240º) (b) tan 675º Solution • –240º is coterminal with 120º. The reference angle is 180º – 120º = 60º. Since –240º lies in quadrant II, the cos(–240º) is negative. • Similarly, tan 675º = tan 315º = –tan 45º = –1.

11. 8.3 Finding Trigonometric Function Values with a Calculator Example Approximate the value of each expression. • cos 49º 12 (b) csc 197.977º Solution Set the calculator in degree mode.

12. 8.3 Finding Angle Measure Example Using Inverse Trigonometric Functions to Find Angles • Use a calculator to find an angle  in degrees that satisfies sin   .9677091705. • Use a calculator to find an angle  in radians that satisfies tan   .25. Solution • With the calculator in degree mode, we findthat an angle having a sine value of .9677091705 is 75.4º. Write this as sin-1 .9677091705  75.4º. • With the calculator in radian mode, we find tan-1 .25  .2449786631.

13. 8.3 Finding Angle Measure Example Find all values of , if  is in the interval [0º, 360º) and Solution Since 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.