Question

solve for the unknown sides of the given triangle. recall the exact values of the trigonometric functions of special angles. notice the right triangle give the trigonometric function of an angle, the leg opposite the angle and the hypotenuse? resources ebook determine length of right triangle side learn by example example video

Explanation:

Step1: Find side $c$ using sine function

We know that $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here $\theta = 30^{\circ}$ and the opposite side to $30^{\circ}$ is $2$. So $\sin30^{\circ}=\frac{2}{c}$. Since $\sin30^{\circ}=\frac{1}{2}$, we have $\frac{1}{2}=\frac{2}{c}$, then $c = 4$.

Step2: Find side $b$ using Pythagorean theorem

In a right - triangle, $a^{2}+b^{2}=c^{2}$. Here $a = 2$ and $c = 4$. So $2^{2}+b^{2}=4^{2}$, which simplifies to $4 + b^{2}=16$. Then $b^{2}=16 - 4=12$, and $b = 2\sqrt{3}$.

Answer:

$b = 2\sqrt{3}$
$c = 4$