Question

Identify the values of the variables. Give your answers in simplest radical form. PLEASE HELP!!

Answer 1

First question:

By definition of sine and cosine, we have

$\sin30^\circ=\dfrac h{\sqrt5}\implies h=\dfrac{\sqrt5}2$

$\cos30^\circ=\dfrac g{\sqrt5}\implies g=\dfrac{\sqrt{15}}2$

Second question:

By definition of tangent, we have

$\tan60^\circ=\dfrac{5\sqrt2}x\implies x=\dfrac{5\sqrt2}{\sqrt3}=\dfrac{5\sqrt6}3$

Then by Pythagoras' theorem,

$(5\sqrt2)^2+x^2=y^2\implies y=\sqrt{(5\sqrt2)^2+\left(\dfrac{5\sqrt6}3\right)^2}=\sqrt{50+\dfrac{50}3}=\sqrt{\dfrac{200}3}=\dfrac{10\sqrt6}3$