Question

find the length of side x in simplest radical form with a rational denominator. find the length of side x in simplest radical form with a rational denominator.

Explanation:

Step1: Identify trigonometric ratio

In a right - triangle, we can use the sine function. $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\sin60^{\circ}=\frac{8}{x}$.
Since $\sin60^{\circ}=\frac{\sqrt{3}}{2}$, we have the equation $\frac{\sqrt{3}}{2}=\frac{8}{x}$.

Step2: Solve for x

Cross - multiply to get $\sqrt{3}x = 16$.
Then $x=\frac{16}{\sqrt{3}}$.

Step3: Rationalize the denominator

Multiply the numerator and denominator by $\sqrt{3}$: $x=\frac{16\sqrt{3}}{3}$.

Answer:

$x = \frac{16\sqrt{3}}{3}$