# 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 can use the identity $\sin(\pi-\theta)=\sin(\theta)$ to find the second solution within $[0,2\pi]$.
$\pi-0.58=2.56$
We use the identity $\sin(\theta+2\pi)=\sin(\theta)$ to extend the two solutions we found to all solutions.
$x=0.58+n\cdot2\pi$
$x=2.56+n\cdot2\pi$
Here, $n$ is any integer.

Problem 1.1
Select one or more expressions that together represent all solutions to the equation.
The answers are in radians. $n$ is any integer.
$\cos(x)=0.15$

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:
\begin{aligned}16\cos(15x)+8&=2\\\\ 16\cos(15x)&=-6\\\\ \cos(15x)&=-0.375\end{aligned}
Use the calculator and round to the nearest thousandth:
$\cos^{-1}(-0.375)=1.955$
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 $15x$):
\begin{aligned} 15x&=1.955+n\cdot2\pi \\\\ x&=\dfrac{1.955+n\cdot2\pi}{15} \\\\ x&=0.130+n\cdot\dfrac{2\pi}{15} \end{aligned}
Similarly, the second solution is $x=-0.130+n\cdot\dfrac{2\pi}{15}$ .

Problem 2.1
Select one or more expressions that together represent all solutions to the equation.
The answers are in radians. $n$ is any integer.
$20\sin(10x)-10=5$
$L(t)$ models the length of each day (in minutes) in Manila, Philippines $t$ days after the spring equinox. Here, $t$ is entered in radians.
$L(t) = {52}\sin\left({\dfrac{2\pi}{365}}t\right) + {728}$
What is the first day after the spring equinox that the day length is $750$ minutes?