1. 6
Inverse Circular Functions and Trigonometric Equations

2. Inverse Circular Functions and Trigonometric Equations6
6.1 Inverse Circular Functions
6.2 Trigonometric Equations I
6.3 Trigonometric Equations II
6.4 Equations Involving Inverse Trigonometric Functions

3. Equations Involving Inverse Trigonometric Functions
6.4
Equations Involving Inverse Trigonometric Functions
Solving for x in Terms of y Using Inverse Functions ▪ Solving Inverse Trigonometric Equations

4. Solve y = 3 cos 2x for x, where x is restricted to the interval
Example 1
SOLVING AN EQUATION FOR A SPECIFIED VARIABLE
Solve y = 3 cos 2x for x, where x is restricted to the interval
Divide by 3.
Definition of arccosine
Multiply by
Because y = 3 cos 2x for x has period π, the restriction ensures that this function is one-to-one and has a one-to-one relationship.

5. Example 2 CHECK Solution set: {1}
SOLVING AN EQUATION INVOLVING AN INVERSE TRIGONOMETRIC FUNCTION
Divide by 2.
Definition of arcsine
CHECK
Solution set: {1}

6. Example 3 and for u in quadrant I,
SOLVING AN EQUATION INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS
and for u in quadrant I,
Substitute.
Definition of arccosine

7. Example 3
SOLVING AN EQUATION INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS (continued)
Sketch u in quadrant I. Use the Pythagorean theorem to find the missing side.

8. Isolate one inverse function on one side of the equation:
Example 4
SOLVING AN INVERSE TRIGONOMETRIC EQUATION USING AN IDENTITY
Isolate one inverse function on one side of the equation:
Definition of arcsine
The arccosine function yields angles in quadrants I and II, so, by definition,

9. From equation (1) and by the definition of the arcsine function,
Example 4
SOLVING AN INVERSE TRIGONOMETRIC EQUATION USING AN IDENTITY (cont.)
Sine sum identity
From equation (1) and by the definition of the arcsine function,
and u lies in quadrant I.

10. From the triangle, we have
Example 4
SOLVING AN INVERSE TRIGONOMETRIC EQUATION USING AN IDENTITY (cont.)
From the triangle, we have
Substitute.
Multiply by 2.
Subtract x.
Square each side.
Distribute, then add 3x2.

11. Now check the solution in the original equation.
Example 4
SOLVING AN INVERSE TRIGONOMETRIC EQUATION USING AN IDENTITY (cont.)
Divide by 4.
Take the square root of each side. Choose the positive root because x > 0.
Now check the solution in the original equation.