Question

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

Explanation:

Step1: Use trigonometric ratio

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\theta = 30^{\circ}\), the side opposite to \(30^{\circ}\) is \(6\), and the hypotenuse is \(x\). So, \(\sin30^{\circ}=\frac{6}{x}\).
Since \(\sin30^{\circ}=\frac{1}{2}\), we have \(\frac{1}{2}=\frac{6}{x}\).

Step2: Solve for \(x\)

Cross - multiply: \(x\times1 = 6\times2\).

Answer:

\(x = 12\)