Chapter 5 Inverse 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

5.1 Inverse sine, cosine, and tangent Inverse sine function Inverse cosine function Inverse tangent function

Inverse Sine Function

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.

Inverse Cosine Function

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.

Inverse Tangent Function

Graphs of the tan and tan-1 Functions

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.

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

Domains for Cotangent, Secant and Cosecant

Graphs of Cotangent, Secant, and Cosecant

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

Identities

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

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.

Suggestions for Solving Trigonometric Equations

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.

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.

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

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.

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