14-2 Solving Trigonometric Equations Using Inverses

Objectives

To evaluate inverse trigonometric functions

To solve trigonometric equations

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You have seen that inverse functions are useful for solving equations. To solve x 3 = 5 , x cubed , equals 5 comma the cube root function gives x = 5 3 . x equals , cube root of 5 , . . To solve x 2 = 5 , x squared , equals 5 comma however, the square root function does not give both solutions 5 square root of 5 and − 5 . negative square root of 5 .

Because the trigonometric functions are periodic, a trigonometric equation like sin θ = 0 sine theta equals 0 has infinitely many solutions. (Think of the sine graph.) Any inverse function for sin θ sine theta must, however, give only one solution.

Essential Understanding To solve some trigonometric equations, you can use an inverse trigonometric function to find one solution. Then you can use periodicity to find all solutions.

Since a function must be single-valued, you define the inverse function for each of sine, cosine, and tangent by inverting only the representative part that has the simplest domain values.

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