# 14.3: Solving Trigonometric Equations

**At Grade**Created by: CK-12

**Objective**

Here you’ll learn how to solve equations with trigonometric functions.

**Review Queue**

Solve the following equations.

1.

2.

3.

## Using Algebra

**Objective**

Here you'll solve trig equations using algebra.

**Guidance**

In the previous concept, we verified trigonometric identities, which are true for every real value of . In this concept, we will solve trigonometric equations. An equation is only true for some values of .

**Example A**

Verify that when .

**Solution:** Substitute in to see if the equations holds true.

This is a true statement, so is a solution to the equation.

**Example B**

Solve .

**Solution:** To solve this equation, we need to isolate and then use inverse to find the values of when the equation is valid. You already did this to find the zeros in the graphing concepts earlier in this chapter.

So, when is the ? Between and . But, the trig functions are periodic, so there are more solutions than just these two. You can write the general solutions as and , where is any integer. You can check your answer graphically by graphing and on the same set of axes. Where the two lines intersect are the solutions.

**Example C**

Solve , where .

**Solution:** In this example, we have an interval where we want to find . Therefore, at the end of the problem, we will need to add or subtract , the period of tangent, to find the correct solutions within our interval.

Using the button on your calculator, we get that . Therefore, we have:

This answer is not within our interval. To find the solutions in the interval, add a couple of times until we have found all of the solutions in .

The two solutions are and 4.4806.

**Guided Practice**

1. Determine if is a solution for .

Solve the following trig equations in the interval .

2.

3.

**Answers**

1. Yes, is a solution.

2. Isolate the and then take the square root of both sides. Don’t forget about the !

The at rad (use your graphing calculator). To find the other value where cosine is positive, subtract 0.243 from , rad.

The at rad, which is in the quadrant. To find the other value where cosine is negative (the quadrant), use the reference angle, 0.243, and add it to . rad.

3. Here, we will find the solution within the given range, .

At this point, we can stop. The range of the cosine function is from 1 to -1. is outside of this range, so there is no solution to this equation.

**Problem Set**

Determine if the following values for . are solutions to the equation .

Solve the following trigonometric equations. If no solutions exist, write *no solution*.

Sole the following trigonometric equations within the interval . If no solutions exist, write *no solution*.

## By Using Quadratic Techniques

**Objective**

Here you'll solve trig equations by factoring and the Quadratic Formula.

**Guidance**

Another way to solve a trig equation is to use factoring or the quadratic formula. Let’s look at a couple of examples.

**Example A**

Solve .

**Solution:** This sine equation looks a lot like the quadratic which factors to be and the solutions are and 1. We can factor the trig equation in the exact same manner. Instead of just , we will have in the factors.

There is no solution for and when .

**Example B**

Solve in the interval .

**Solution:** To solve this equation, use the Pythagorean Identity . Solve for either cosine and substitute into the equation.

Solving each factor for , we get and and .

**Example C**

Solve in the interval .

**Solution:** This equation is not factorable so you have to use the Quadratic Formula.

and

The first answer is within the range, but the second is not. To adjust -0.954 to be within the range, we need to find the answer in the second quadrant, .

**Guided Practice**

Solve the following trig equations using any method in the interval .

1.

2.

3.

**Answers**

1. Put everything onto one side of the equation and factor out a cosine.

2. Recall that from the Negative Angle Identities.

3. This quadratic is not factorable, so use the quadratic formula.

The other solutions in the range are and .

**Problem Set**

Solve the following trig equations using any method. Find all solutions in the interval . Round any decimal answers to 4 decimal places.

Using your graphing calculator, graph the following equations and determine the points of intersection in the interval .

- .

- .