2. Copyright © 2009 Pearson Education, Inc. CHAPTER 7: Trigonometric Identities, Inverse Functions, and Equations 7.1Identities: Pythagorean and Sum and Difference 7.2Identities: Cofunction, Double-Angle, and Half- Angle 7.3Proving Trigonometric Identities 7.4Inverses of the Trigonometric Functions 7.5Solving Trigonometric Equations

3. Copyright © 2009 Pearson Education, Inc. 7.4 Inverses of the Trigonometric Functions  Find values of the inverse trigonometric functions.  Simplify expressions such as sin (sin -1 x) and sin -1 (sin x).  Simplify expressions involving composition such as sin (cos –1 1/2) without using a calculator.  Simplify expressions such as sin arctan (a/b) by making a drawing and reading off appropriate ratios.

4. Slide 7.4 - 4 Copyright © 2009 Pearson Education, Inc. Inverse Sine Function The graphs of an equation and its inverse are reflections of each other across the line y = x. However, the inverse is not a function as it is drawn.

5. Slide 7.4 - 5 Copyright © 2009 Pearson Education, Inc. Inverse Sine Function We must restrict the domain of the inverse sine function. It is fairly standard to restrict it as shown here. The domain is [–1, 1]. The range is [–π/2, π/2].

6. Slide 7.4 - 6 Copyright © 2009 Pearson Education, Inc. Inverse Cosine Function The graphs of an equation and its inverse are reflections of each other across the line y = x. However, the inverse is not a function as it is drawn.

7. Slide 7.4 - 7 Copyright © 2009 Pearson Education, Inc. Inverse Cosine Function We must restrict the domain of the inverse cosine function. It is fairly standard to restrict it as shown here. The domain is [–1, 1]. The range is [0, π].

8. Slide 7.4 - 8 Copyright © 2009 Pearson Education, Inc. Inverse Tangent Function The graphs of an equation and its inverse are reflections of each other across the line y = x. However, the inverse is not a function as it is drawn.

9. Slide 7.4 - 9 Copyright © 2009 Pearson Education, Inc. Inverse Sine Function We must restrict the domain of the inverse tangent function. It is fairly standard to restrict it as shown here. The domain is (–∞, ∞). The range is (–π/2, π/2).

10. Slide 7.4 - 10 Copyright © 2009 Pearson Education, Inc. Inverse Trigonometric Functions FunctionDomainRange

11. Slide 7.4 - 11 Copyright © 2009 Pearson Education, Inc. Graphs of the Inverse Trigonometric Functions

12. Slide 7.4 - 12 Copyright © 2009 Pearson Education, Inc. Graphs of the Inverse Trigonometric Functions

13. Slide 7.4 - 13 Copyright © 2009 Pearson Education, Inc. Example Find each of the following function values. Find  such that sin  =. In the restricted range [–π/2, π/2], the only number with sine of is π/4. Solution:

14. Slide 7.4 - 14 Copyright © 2009 Pearson Education, Inc. Example Solution continued: Find  such that cos  = –1/2. In the restricted range [0, π], the only number with cosine of –1/2 is 2π/3.

15. Slide 7.4 - 15 Copyright © 2009 Pearson Education, Inc. Example Solution continued: Find  such that tan  = In the restricted range (–π/2, π/2), the only number with tangent of is –π/6.

16. Slide 7.4 - 16 Copyright © 2009 Pearson Education, Inc. Example Approximate the following function value in both radians and degrees. Round radian measure to four decimal places and degree measure to the nearest tenth of a degree. Solution: Press the following keys (radian mode): Readout: Rounded: 1.8430 Rounded: 105.5º Change to degree mode and press the same keys: Readout:

17. Slide 7.4 - 17 Copyright © 2009 Pearson Education, Inc. Composition of Trigonometric Functions for all x in the domain of sin –1 for all x in the domain of cos –1 for all x in the domain of tan –1

18. Slide 7.4 - 18 Copyright © 2009 Pearson Education, Inc. Example Simplify each of the following. Solution: a) Since is in the domain, [–1, 1], it follows that b) Since 1.8 is not in the domain, [–1, 1], we cannot evaluate the expression. There is no number with sine of 1.8. So, sin (sin –1 1.8) does not exist.

19. Slide 7.4 - 19 Copyright © 2009 Pearson Education, Inc. Special Cases for all x in the range of sin –1 for all x in the range of cos –1 for all x in the range of tan –1

20. Slide 7.4 - 20 Copyright © 2009 Pearson Education, Inc. Example Simplify each of the following. Solution: a) Since π/6 is in the range, (–π/2, π/2), it follows that b) Since 3π/4 is not in the range, [–π/2, π/2], we cannot apply sin –1 (sin x) = x.

21. Slide 7.4 - 21 Copyright © 2009 Pearson Education, Inc. Example Find Solution: cot –1 is defined in (0, π), so consider quadrants I and II. Draw right triangles with legs x and 2, so cot  = x/2.