Algebra and Trigonometry, 5th Edition

7.5 Solving Trigonometric Equations

• Solve trigonometric equations.

When an equation contains a trigonometric expression with a variable, such as $\text{cos}x$, it is called a trigonometric equation. Some trigonometric equations are identities, such as ${\text{sin}}^{2}x+{\text{cos}}^{2}x=1$. Now we consider equations, such as $2\text{cos}x=-1$, that are usually not identities. As we have done for other types of equations, we will solve such equations by finding all values for $x$ that make the equation true.

JUST IN TIME 14

Example 1

Solve:$2\text{cos}x=-1$.

Algebraic Solution

We first solve for cos $x$:

$$\begin{array}{ccc}\hfill 2\text{cos}x& =\hfill & -1\hfill \\ \hfill \text{cos}x& =\hfill & -{\displaystyle \frac{1}{2}}\text{.}\hfill \end{array}$$

The solutions are numbers that have a cosine of $-{\displaystyle \frac{1}{2}}$. To find them, we use the unit circle (see Section 6.5).

There are only two points on the unit circle for ...