Question

find the length of side x in simplest radical form with a rational denominator.
(there is a right triangle with angles 30°, 60°, and 90°, one leg is 4, the other leg is x)

Explanation:

Step1: Identify trigonometric ratio

We use tangent: $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. For the $60^\circ$ angle, opposite side is $x$, adjacent is $4$.
$\tan(60^\circ)=\frac{x}{4}$

Step2: Substitute $\tan(60^\circ)$ value

$\tan(60^\circ)=\sqrt{3}$, so:
$\sqrt{3}=\frac{x}{4}$

Step3: Solve for $x$

Multiply both sides by 4:
$x=4\sqrt{3}$

Answer:

$x=4\sqrt{3}$