Solving a trigonometric equation is just like solving a regular equation. You will use factoring and other algebraic techniques to get the variable on one side. The biggest difference with trigonometric equations is the opportunity for there to be an infinite number of solutions that must be described with a pattern. The equation

### Solving Trigonometric Equations

The identities you have learned are helpful in solving **trigonometric equations**. The goal of solving an equation hasn’t changed. Do whatever it takes to get the variable alone on one side of the equation. Factoring, especially with the Pythagorean identity, is critical.

When solving trigonometric equations, try to give exact (non-rounded) answers. If you are working with a calculator, keep in mind that while some newer calculators can provide exact answers like

When solving, if the two sides of the equation are always equal, then the equation is an identity. If the two sides of an equation are never equal, as with

### Examples

#### Example 1

Earlier, you were asked how you could describe the many solutions of

#### Example 2

Solve the following equation algebraically and confirm graphically on the interval

Solving the first part set equal to zero within the interval yields:

Solving the second part set equal to zero yields:

These are the six solutions that will appear as intersections of the two graphs

Note that the terms “general solution,” “completely solve”, and“solve exactly”

#### Example 3

Determine the general solution to the following equation.

One solution is

Notice that trigonometric equations may have an infinite number of solutions that repeat in a certain pattern because they are periodic functions. When you see these directions remember to find all the solutions by using notation like in this example.

#### Example 4

Solve the following equation.

This equation is always true which means the right side is always equal to the left side. This is an identity.

#### Example 5

Solve the following equation exactly.

Start by factoring:

Note that

These are the solutions within the interval

### Review

Solve each equation on the interval

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2.

Find approximate solutions to each equation on the interval

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7.

Solve each equation on the interval \begin{align*}[0,360^\circ)\end{align*}.

8. \begin{align*}2 \sin \left(x-\frac{\pi}{2}\right)=1\end{align*}

9. \begin{align*}4 \cos (x-\pi)=4\end{align*}

Solve each equation on the interval \begin{align*}[2 \pi, 4 \pi)\end{align*}.

10. \begin{align*}\cos^2 x+2 \cos x+1=0\end{align*}

11. \begin{align*}3 \sin x=2 \cos^2 x\end{align*}

12. \begin{align*}\tan x \sin^2 x=\tan x\end{align*}

13. \begin{align*}\sin^2 x+1=2 \sin x\end{align*}

14. \begin{align*}\sec^2 x=4\end{align*}

15. \begin{align*}\sin^2 x-4=\cos^2 x-\cos 2x-4\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.5.