Chapter 5 Inverse Trigonometric Functions; Trigonometric Equations and Inequalities

1. Chapter 5Inverse Trigonometric Functions;Trigonometric Equations and Inequalities 5.1 Inverse sine, cosine, and tangent 5.2 Inverse cotangent, secant, and cosecant 5.3 Trigonometric Equations: An Algebraic Approach 5.4 Trigonometric Equations: A Graphing Calculator Approach

2. 5.1 Inverse sine, cosine, and tangent • Inverse sine function • Inverse cosine function • Inverse tangent function

3. Inverse Sine Function

4. Finding the Exact Value of sin-1 x • Example: Find the exact value of sin-1 (√3/2) • Solution: y = sin-1 (√3/2) is equivalent to sin y = √3/2. Find the value of y that lies between –p/2 and p/2 on the unit circle. • The answer is p/3.

5. Inverse Cosine Function

6. Finding the Exact Value of cos-1x • Example: Find the exact value of cos-1 ½. • Solution: • y = cos-1 ½ is equivalent to cos y = ½. We find the value of y on the unit circle between 0 and p for which this is true. • The answer is p/3.

7. Inverse Tangent Function

8. Graphs of the tan and tan-1 Functions

9. Finding the Exact Value of tan-1 x • Example: Find the exact value of tan-1 (-1/√3). • Solution: • Y = tan-1 (-1/√3) is equivalent to tan y = -1/√3. Find the value of y on the unit circle between –p/2 and p/2 for which this is true. • Answer is –p/6.

10. 5.2 Inverse Cotangent, Secant, and Cosecant Functions • Definition of inverse cotangent, secant, and cosecant functions • Calculator evaluation

11. Domains for Cotangent, Secant and Cosecant

12. Graphs of Cotangent, Secant, and Cosecant

13. Finding the Exact Value of arccot (-1) • Example: Find the exact value of arccot (-1) • Solution: • y = arccot(-1) is equivalent to cot y = -1. Find the value of y on the unit circle between 0 and p that makes this true. • The answer is 3p/4

14. Identities

15. 5.3 Trigonometric Equations:An Algebraic Approach • Introduction • Solving trigonometric equations using an algebraic approach

16. Solving a Simple Sine Equation • Find all solutions in the unit circle to sin x = 1/√2. • Solution: • Use the unit circle to determine that one solution is x = p/4. • It can be seen that another point on the circle with the desired height is x = 3p/4.

17. Suggestions for Solving Trigonometric Equations

18. Exact Solutions Using Factoring • Example: Find all solutions in [0, 2p] to 2 sin2x + sin x = 0 • Solution: • 2 sin2x + sin x = 0 • sin x(2 sin x + 1) = 0 • sin x = 0 or sin x = -1/2 • Find these values on the unit circle. • The solutions are x = 0, p, 7p/6, and 11p/6.

19. Exact Solutions Using Identities and Factoring • Example: Find all solutions for sin 2x = sin x, 0  x  2p. • Solution: • sin 2x = sin x • 2 sin x cos x = sin x • 2 sin x cos x – sin x = 0 • sin x (2 cos x – 1) = 0 • sin x = 0 or cos x = ½ • From the unit circle we find 4 solutions: x = 0, p/3, p, and 5p/3.

20. 5.4 Trigonometric Equations and Inequalities: A Graphing Calculator Approach • Solving trigonometric equations using a graphing calculator • Solving trigonometric inequalities using a graphing calculator

21. Solutions Using a Graphing Calculator • Example: Graph y1=sin(x/2) and y2= 0.2x – 0.5 over [-4p, 4p]. • Use the INTERSECT command to find that x=5.1609 is the intersection. • Use the ZOOM command to find that there is no intersection in the third quadrant.

22. Solution Using a Graphing Calculator • Example: Find all real solutions (to four decimal places) to tan(x/2) = 5x – x2 over [0, 3p]. • Graph y = tan(x/2) and y = 5x – x2 over 0X3p and -10Y10. • Use the INTERSECT command to find three solutions: x = 0.0000, 2.8292, 5.1272