Question

Find the value of both variables. Answer must be in simplest radical form.

Answer 1

Statement Problem: Find the missing variables in the figure;

Solution:

Recall the trigonometry ratios;

$\begin{gathered} \sin \theta=\frac{opposite}{hypotenuse} \\ \cos \theta=\frac{\text{adjacent}}{hypotenuse} \\ \tan \theta=\frac{opposite}{\text{adjacent}} \end{gathered}$

In this case, the adjacent sides of the angle is given.

The hypotenuse side is y. Thus, we would apply the cosine, we have;

$\begin{gathered} \cos 60^o=\frac{5\sqrt[]{7}}{y} \\ \frac{1}{2}=\frac{5\sqrt[]{7}}{y} \\ \text{cross}-m\text{ultiply, we have;} \\ y=2\times5\sqrt[]{7} \\ y=10\sqrt[]{7} \end{gathered}$

Also, we would apply the pythagoras theorem to find the opposite side given as x.

The pythagoras theorem is;

$\begin{gathered} h^2=o^2+a^2 \\ o^2=h^2-a^2 \\ a^2=h^2-o^2 \\ \text{Where h=hypotenuse, a=adjacent, o=opposite} \end{gathered}$

Thus, the opposite side, x is;

$\begin{gathered} x^2=(10\sqrt[]{7})^2-(5\sqrt[]{7})^2 \\ x^2=700-175 \\ x^2=525 \\ x=\sqrt[]{525} \\ x=5\sqrt[]{21} \end{gathered}$