Chapter 5Inverse Trigonometric Functions;

Trigonometric Equations and Inequalities

5.1 Inverse sine, cosine, and tangent 5.2 Inverse cotangent, secant, and cosecant5.3 Trigonometric Equations: An Algebraic Approach5.4 Trigonometric Equations: A Graphing Calculator Approach

5.1 Inverse sine, cosine, and tangent

Inverse sine functionInverse cosine functionInverse 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 –/2 and /2 on the unit circle.

The answer is /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 for which this is true.

The answer is /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 –/2 and /2 for which this is true.

Answer is –/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 that makes this true.

The answer is 3/4

Identities

5.3 Trigonometric Equations:An Algebraic Approach

IntroductionSolving 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 = /4.

It can be seen that another point on the circle with the desired height is

x = 3/4.

Suggestions for Solving Trigonometric Equations

Exact Solutions Using Factoring

Example: Find all solutions in [0, 2] to 2 sin2x + sin x = 0

Solution:2 sin2x + sin x = 0sin x(2 sin x + 1) = 0sin x = 0 or sin x = -1/2Find these values on the unit

circle.The solutions are x = 0, ,

7/6, and 11/6.

Exact Solutions Using Identities and Factoring

Example: Find all solutions for sin 2x = sin x, 0 x 2.

Solution:sin 2x = sin x2 sin x cos x = sin x2 sin x cos x – sin x = 0sin x (2 cos x – 1) = 0sin x = 0 or cos x = ½From the unit circle we find 4

solutions: x = 0, /3, , and 5/3.

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 [-4, 4].

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, 3].

Graph y = tan(x/2) and y = 5x – x2 over 0X3 and -10Y10.

Use the INTERSECT command to find three solutions:

x = 0.0000, 2.8292, 5.1272