Question

part b what are the exact side lengths of the triangle shown? 21 cm

Explanation:

Step1: Use sine - ratio for side a

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. For the $30^{\circ}$ angle, $\sin30^{\circ}=\frac{a}{21}$. Since $\sin30^{\circ}=\frac{1}{2}$, we have $\frac{a}{21}=\frac{1}{2}$.
$a = 21\times\sin30^{\circ}=21\times\frac{1}{2}=\frac{21}{2}$ cm.

Step2: Use cosine - ratio for side c

For the $30^{\circ}$ angle, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. So $\cos30^{\circ}=\frac{c}{21}$. Since $\cos30^{\circ}=\frac{\sqrt{3}}{2}$, we get $c = 21\times\cos30^{\circ}=21\times\frac{\sqrt{3}}{2}=\frac{21\sqrt{3}}{2}$ cm.

Answer:

$a=\frac{21}{2}$
$c=\frac{21\sqrt{3}}{2}$