Question

1. the right triangle has the listed measures. what is the length of the opposite side of the triangle?

triangle image: vertical side labeled x, horizontal side labeled 15 in, angle at bottom right labeled 30 degrees

Explanation:

Step1: Identify trigonometric ratio

We use the tangent function, which is $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$. Here, $\theta = 30^\circ$, adjacent side is $15$ in, and opposite side is $x$. So $\tan(30^\circ) = \frac{x}{15}$.

Step2: Solve for \( x \)

We know that $\tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.577$. Multiply both sides by $15$: $x = 15 \times \tan(30^\circ) = 15 \times \frac{1}{\sqrt{3}} = \frac{15}{\sqrt{3}}$. Rationalize the denominator: $\frac{15\sqrt{3}}{3} = 5\sqrt{3} \approx 8.66$ in.

Answer:

The length of the opposite side is $5\sqrt{3}$ inches (or approximately $8.66$ inches).