# Right triangle trigonometry review

CCSS Math: HSG.SRT.C.8
Review right triangle trigonometry and how to use it to solve problems.

## What are the basic trigonometric ratios?

$\large\sin(\angle A)=$$\large\dfrac{\blueD{\text{opposite}}}{\goldD{\text{hypotenuse}}}$
$\large\cos(\angle A)=$$\large\dfrac{\purpleC{\text{adjacent}}}{\goldD{\text{hypotenuse}}}$
$\large\tan(\angle A)=$$\large\dfrac{\blueD{\text{opposite}}}{\purpleC{\text{adjacent}}}$

## Practice set 1: Solving for a side

Trigonometry can be used to find a missing side length in a right triangle. Let's find, for example, the measure of $AC$ in this triangle:
We are given the measure of angle $\angle B$ and the length of the $\goldD{\text{hypotenuse}}$, and we are asked to find the side $\blueD{\text{opposite}}$ to $\angle B$. The trigonometric ratio that contains both of those sides is the sine:
\begin{aligned} \sin(\angle B)&=\dfrac{\blueD{AC}}{\goldD{AB}} \\\\ \sin(40^\circ)&=\dfrac{AC}{7}\quad\gray{\angle B=40^\circ, AB=7} \\\\ 7\cdot\sin(40^\circ)&=AC \end{aligned}
Now we evaluate using the calculator and round:
$AC=7\cdot\sin(40^\circ)\approx 4.5$
Problem 1.1
$BC=$

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

## Practice set 2: Solving for an angle

Trigonometry can also be used to find missing angle measures. Let's find, for example, the measure of $\angle A$ in this triangle:
We are given the length of the side $\purpleC{\text{adjacent}}$ to the missing angle, and the length of the $\goldD{\text{hypotenuse}}$. The trigonometric ratio that contains both of those sides is the cosine:
\begin{aligned} \cos(\angle A)&=\dfrac{\purpleC{AC}}{\goldD{AB}} \\\\ \cos(\angle A)&=\dfrac{6}{8}\quad\gray{AC=6, AB=8} \\\\ \angle A&=\cos^{-1}\left(\dfrac{6}{8}\right) \end{aligned}
Now we evaluate using the calculator and round:
$\angle A=\cos^{-1}\left(\dfrac{6}{8}\right) \approx 41.41^\circ$
Problem 2.1
$\angle A=$
$^\circ$
Howard is designing a chair swing ride. The swing ropes are $5$ meters long, and in full swing they tilt in an angle of $29^\circ$. Howard wants the chairs to be $2.75$ meters above the ground in full swing.