Question

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

Explanation:

Step1: Use trigonometric ratio

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

Step2: Verify using another trigonometric ratio (optional)

We can also use \(\cos60^{\circ}\). In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). For \(\theta = 60^{\circ}\), the adjacent side is \(x\) and hypotenuse is \(1\).
\(\cos60^{\circ}=\frac{x}{1}\)
Since \(\cos60^{\circ}=\frac{1}{2}\), we get \(x=\frac{1}{2}\)

Answer:

\(x=\frac{1}{2}\)