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Trigonometry5.3 Trigonometric Equations

Objective: Use standard algebraic techniques like collecting like terms and factoring to solve trigonometric equations.

Example 1: Solving a Trig Equation

Step 1: Isolate the trig function on one side of the equation.

Step 2: Use an inverse function.

Step 3: Since sine and cosine have a period of 2π, the solutions repeat every 2πn intervals so express your answer to show this.

Example 2: Collect Like Terms

Step 1: Isolate the trig function on one side of the equation.

Step 2: Use an inverse function and find all possible answers on the specified interval.

Example 3: Extracting Square Roots

Step 1: Isolate the trig function on one side of the equation.

Step 2: Use an inverse function and find all possible answers on the specified interval.

2 cos x−1=0

tan x+√3=0 , [ 0,2π )

4 sin2 x−3=0 , [0,2π )

Example 4: Quadratic Types

Step 1: If an equation looks quadratic, factor it.

Step 2: Set each factor equal to zero.

Step 3: Use an inverse function and find all possible answers on the specified interval. Keep in mind that sine and cosine must be less than 1.

Example 5: Rewriting as one function.

Step 1: Use a Pythagorean Identity and rewrite sec in terms of tan so the entire function is in terms of tan.

Step 2: Factor.

Step 3: Use an inverse function.

Step 3: Since tan has a period of π, the solutions repeat every πn intervals so express your answer to show this.

2 cos2 x−cos x−1=0 , [ 0, 2π )

3 sec2 x−2 tan2 x−4=0

Example 6: Squaring/Converting

Step 1: Since the equation is not factorable and you cannot write in terms of one function, try squaring both sides.

Step 2: Factor out the greatest common factor.

Step 3: Use an inverse function and find all possible answers on the specified interval. Keep in mind that sine and cosine must be less than 1.

Example 7: You Try It!

Homework: Page 376, #25, 36, 37, 39, 49, 51, 53

sin x+1=cos x , [ 0, 2π )

4 cos2 x−1=0