Solving trig equations is an important process in mathematics. Quite often you'll see powers of trigonometric functions and be asked to solve for the values of the variable which make the equation true. For example, suppose you were given the trig equation

### Trigonometric Equations Using Factoring

You have no doubt had experience with factoring. You have probably factored equations when looking for the possible values of some variable, such as "x". It might interest you to find out that you can use the same factoring method for more than just a variable that is a number. You can factor trigonometric equations to find the possible values the function can take to satisfy an equation.

Algebraic skills like factoring and substitution that are used to solve various equations are very useful when solving trigonometric equations. As with algebraic expressions, one must be careful to avoid dividing by zero during these maneuvers.

#### Solving for Unknown Values

1. Solve

2. Solve

Pull out

There is a common factor of

Think of the

3. Solve

Some trigonometric equations have no solutions. This means that there is no replacement for the variable that will result in a true expression.

### Examples

#### Example 1

Earlier, you were asked to solve this:

Subtract

Now set each factor equal to zero and solve. The first is

And now for the other term:

#### Example 2

Solve the trigonometric equation

#### Example 3

Solve

#### Example 4

Find all the solutions for the trigonometric equation

### Review

Solve each equation for

cos2(x)+2cos(x)+1=0 1−2sin(x)+sin2(x)=0 2cos(x)sin(x)−cos(x)=0 sin(x)tan2(x)−sin(x)=0 sec2(x)=4 sin2(x)−2sin(x)=0 3sin(x)=2cos2(x) 2sin2(x)+3sin(x)=2 tan(x)sin2(x)=tan(x) 2sin2(x)+sin(x)=1 2cos(x)tan(x)−tan(x)=0 sin2(x)+sin(x)=2 - \begin{align*}\tan(x)(2\cos^2(x)+3\cos(x)-2)=0\end{align*}
- \begin{align*}\sin^2(x)+1=2\sin(x)\end{align*}
- \begin{align*}2\cos^2(x)-3\cos(x)=2\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.4.