Question

what are the exact side lengths of the triangle shown? 30° b a 60° 8 mm mm b = mm

Explanation:

Step1: Use sine - cosine relations

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$ and $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Given the side of length 8 mm, and angles 30° and 60°. If the side of length 8 mm is adjacent to the 30° angle.
For side $a$ (hypotenuse), $\cos30^{\circ}=\frac{8}{a}$. Since $\cos30^{\circ}=\frac{\sqrt{3}}{2}$, we have $a=\frac{8}{\cos30^{\circ}}$.

Step2: Calculate side $a$

$a = \frac{8}{\frac{\sqrt{3}}{2}}=\frac{16\sqrt{3}}{3}\text{ mm}$.

Step3: Calculate side $b$

For side $b$ (opposite to the 30° angle), $\tan30^{\circ}=\frac{b}{8}$. Since $\tan30^{\circ}=\frac{1}{\sqrt{3}}$, then $b = \frac{8}{\sqrt{3}}=\frac{8\sqrt{3}}{3}\text{ mm}$.

Answer:

$a=\frac{16\sqrt{3}}{3}$, $b = \frac{8\sqrt{3}}{3}$