Trigonometric Equations

OBJECTIVES:Use standard algebraic techniques to solve

trigonometric equationsSolve trigonometric equations of quadratic

type

IntroductionO To solve a trigonometric equation,

use standard algebraic techniques (when possible) such as collecting like terms and factoring. Your preliminary goal in solving a trigonometric equation is to isolate the trigonometric function on one side of the equation.

Solve the trigonometric equation

O To solve the equation for x

O These solutions are in the interval

• But there are infinitely many other solutions

2sin 1 0

2sin 1

1sin

27 11

,6 6

x

x

x

x

0,2

7 112 and 2

6 6x n x n

EX 1: SolveO A) sin 2 sinx x

EX 1: SolveO B) 23tan 1 0x

EX 1: SolveO C) 22cos cos 1 0x x

EX 1: SolveO D) 2tan tan 2 0x x

Solving EquationO The equations in Example 1 involved

only one trigonometric function. When two or more functions occur in the same equation, collect all terms on one side and try to separate the functions by factoring, using the Quadratic Formula or by using appropriate identities. This may produce factors that yield no solutions.

EX 2: SolveO A) sin tan sinx x x

EX 2: SolveO B) 22sin 3cos 3 0x x

EX 2: SolveO C) 2cot 3cot 1x x

Solving EquationsO Sometimes you must square each

side of an equation to obtain a quadratic. Because this procedure can introduce extraneous solutions, you should check any solutions in the original equation to see whether they are valid or extraneous.

EX 3: SolveO A) cos 1 sinx x

EX 2: SolveO B) tan 3 secx x