Question

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

Explanation:

Step1: Use cosine function

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here \(\theta = 30^{\circ}\), adjacent side is \(x\) and hypotenuse is \(4\). So \(\cos30^{\circ}=\frac{x}{4}\).

Step2: Substitute the value of \(\cos30^{\circ}\)

Since \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\), we have \(\frac{\sqrt{3}}{2}=\frac{x}{4}\).

Step3: Solve for \(x\)

Cross - multiply: \(x = 4\times\frac{\sqrt{3}}{2}\). Simplify: \(x = 2\sqrt{3}\).

Answer:

\(2\sqrt{3}\)