# Trigonometric equations review

Review your trigonometric equation skills by solving a sequence of equations in increasing complexity.

## Practice set 1: Basic equations

### Example: Solving $\sin(x)=0.55$

Let's use the calculator and round to the nearest hundredth.

$\sin^{-1}(0.55)=0.58$

*(We are using radians.)*We can use the identity $\sin(\pi-\theta)=\sin(\theta)$ to find the second solution within $[0,2\pi]$.

We use the identity $\sin(\theta+2\pi)=\sin(\theta)$ to extend the two solutions we found to all solutions.

$x=0.58+n\cdot2\pi$

$x=2.56+n\cdot2\pi$

$x=2.56+n\cdot2\pi$

Here, $n$ is any integer.

### Check your understanding

*Want to try more problems like this? Check out this exercise.*

## Practice set 2: Advanced equations

### Example: Solving $16\cos(15x)+8=2$

First, we isolate the trig expression:

Use the calculator and round to the nearest thousandth:

Use the identity $\cos(\theta)=\cos(-\theta)$ to find that the second solution within $[-\pi,\pi]$ is $-1.955$.

Use the identity $\cos(\theta)=\cos(\theta+2\pi)$ to find all the solutions to our equation from the two angles we found above. Then we solve for $x$ (remember that our argument is $15x$):

Similarly, the second solution is $x=-0.130+n\cdot\dfrac{2\pi}{15}$ .

### Check your understanding

*Want to try more problems like this? Check out this exercise.*