**SOLVING SIMPLE TRIGONOMETRIC
EQUATIONS
Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to
ingrid.stewart@csn.edu.**

Learning Objectives - This is what you must know after studying the lecture and doing the practice problems!

1.Solve simple trigonometric equations on a restricted solution interval..

2. Find ALL solutions of simple trigonometric equations

In this lesson, we will learn how to solve simple trigonometric equations of the following forms:

asin(bx) = C

acos(bx) = C

atan(bx) = CNote that

andb = 1andaare real numbers withCnot equal to 0.aAll trigonometric equations have infinitely many solutions! Often, the solutions are restricted to an interval in which case we may have zero to several solutions. Other times we are asked to find all solutions.

Solving Simple Trigonometric Equations on a Restricted Solution Interval- see #1, 2, 4 through 9in the "Examples" document

Step 1:Examine the solution interval and keep it in mind. If necessary, isolate the trigonometric ratio. "Isolate" means that the trigonometric ratio must be the only term on one side of the equal sign and its coefficient must be 1.

Step 2:Use the

Inverse Trigonometric Functionconcept to find a solution for the unknown angle.HINT: If you are asked to find the EXACT radian solutions, it might be easier to convert to degrees!

Step 3:If necessary, find the

reference anglefor the solution in Step 2. Areference anglewill not exist if the solution in Step 2 is aquadrantal angle.

Step 4:Find the solution(s) on the given solution interval.

1. Examine the sign of the trigonometric ratio in Step 1. Is it positive or negative?

2a. If a

reference angleexists in Step 3 and knowing the sign of the trigonometric ratio from Step 1, use this in conjuction withAll Students Take Calculusto find the solution(s) on the given solution interval!2b. If a

reference angledoes not exist in Step 3, we must know the values of the trigonometric ratios ofquadrantal anglesto find the solution(s) on the given solution interval.

Finding ALL Solutions of Simple Trigonometric Equations- see #3, 10 through 12in the "Examples" documentSometimes, we are not given a solution interval. Instead, we are asked to find ALL solutions. In that case, we do the following:

- Find the solutions on the interval between
and or between0and0^{o}.360^{o}- If the solutions in (1) are more than or
apart, add180^{o}or2kto each of these solutions, where360^{o}kis any integer.k- If the solutions in (1) are exactly or
apart, add180^{o}orkto the smallest solution.180k^{o}