Question

Find the values of x and y in the following right triangle. Enter square roots not decimals.

Answer 1

Recall the following trigonometric identities. If the legs of the right triangle have lengths a and b, the hypotenuse has length c, and the side a is adjacent to an angle θ, then:

$\begin{gathered} \sin \theta=\frac{b}{c} \\ \cos \theta=\frac{a}{c} \end{gathered}$

Then, for the given right triangle:

$\begin{gathered} \sin (30º)=\frac{x}{8} \\ \cos (30º)=\frac{y}{8} \end{gathered}$

Then, x and y are given by the expressions:

$\begin{gathered} x=8\cdot\sin (30º)=8\cdot\frac{1}{2}=4 \\ y=8\cdot\cos (30º)=8\cdot\frac{\sqrt[]{3}}{2}=4\cdot\sqrt[]{3} \end{gathered}$

Therefore, the answers are:

$\begin{gathered} x=4\cdot\sqrt[]{3} \\ y=4 \end{gathered}$