Question

Find the missing sides of the triangle. Leave youranswers as simplified radicals.

Answer 1

Explanation:

Consider the following right triangle:

To find the missing sides x and y, we can apply the following trigonometric ratios:

$\cos(60^{\circ})=\frac{adjacent\text{ side to the angle 60}^{\circ}}{Hypotenuse}$

$\sin(60^{\circ})=\frac{opposite\text{ side to the angle 60}^{\circ}}{Hypotenuse}$

and

$\tan(60^{\circ})=\frac{opposite\text{ side to the angle 60}^{\circ}}{adjacent\text{ side to the angle 60}^{\circ}}$

thus, applying the data of the problem to the last equation, we get:

$\tan(60^{\circ})=\frac{opposite\text{ side to the angle 60}^{\circ}}{adjacent\text{ side to the angle 60}^{\circ}}=\frac{15}{y}$

that is:

$\tan(60^{\circ})=\frac{15}{y}$

solving for y, we obtain:

$y=\frac{15}{\tan(60^{\circ})}=\frac{15}{\sqrt{3}}$

On the other hand, applying the above data to the first equation, we get:

$\cos(60^{\circ})=\frac{adjacent\text{ side to the angle 60}^{\circ}}{Hypotenuse}=\frac{y}{x}=\frac{15}{\sqrt{3}}\text{ }\cdot\frac{1}{x}$

or

$\cos(60^{\circ})=\frac{15}{\sqrt{3}\text{ x}}\text{ }$

solving for x, we obtain:

$x=\frac{15}{\sqrt{3}\cdot\cos(60)}=\text{ }\frac{15}{\sqrt{3\text{ }}\cdot1/2}=\frac{2(15)}{\sqrt{3}}=\frac{30}{\sqrt{3}}$

we can conclude that the correct answer is:

Answer:

$x=\frac{30}{\sqrt{3}}$

and

$y=\frac{15}{\sqrt{3}}$