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6Inverse Circular Functions and Trigonometric Equations

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Equations Involving Inverse Trigonometric Functions

6.4

Solution for x in Terms of y Using Inverse Functions ▪ Solving Inverse Trigonometric Equations

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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

0, .2π⎡ ⎤

⎢ ⎥⎣ ⎦

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.

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Example 2 SOLVING AN EQUATION INVOLVING AN INVERSE TRIGONOMETRIC FUNCTION

CHECK

Solution set: {1}

Divide by 2.

Definition of arcsine

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Example 3 SOLVING AN EQUATION INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS

and for u in quadrant I,

Substitute.

Definition of arccosine

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Example 3 SOLVING AN EQUATION INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS (continued)

Sketch u in quadrant I. Use the Pythagorean theorem to find the missing side.

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Example 4 SOLVING AN INVERSE TRIGONOMETRIC EQUATION USING AN IDENTITY

Isolate one inverse function on one side of the equation:

The arccosine function yields angles in quadrants I and II, so, by definition,

Definition of arcsine

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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.

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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.

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Example 4 SOLVING AN INVERSE TRIGONOMETRIC EQUATION USING AN IDENTITY (cont.)

Take the square root of each side. Choose the positive root because x > 0.

Now check the solution in the original equation.

Divide by 4.