Question

solve for the unknown sides of the given triangle. recall the exact values of the trigonometric functions of special angles. notice the right 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: Use sine function for side a

We know that $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here $\theta = 60^{\circ}$ and hypotenuse is 16. So $\sin60^{\circ}=\frac{a}{16}$. Since $\sin60^{\circ}=\frac{\sqrt{3}}{2}$, we have $\frac{\sqrt{3}}{2}=\frac{a}{16}$.
$a = 16\times\frac{\sqrt{3}}{2}=8\sqrt{3}$

Step2: Use cosine function for side c

We know that $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here $\theta = 60^{\circ}$ and hypotenuse is 16. So $\cos60^{\circ}=\frac{c}{16}$. Since $\cos60^{\circ}=\frac{1}{2}$, we have $\frac{1}{2}=\frac{c}{16}$.
$c = 16\times\frac{1}{2}=8$

Answer:

$a = 8\sqrt{3}$, $c = 8$