Question

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

Explanation:

Step1: Use trigonometric ratio

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here \(\theta = 30^{\circ}\), adjacent side \(= 4\), hypotenuse \(=x\). So \(\cos30^{\circ}=\frac{4}{x}\).
Since \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\), we have \(\frac{\sqrt{3}}{2}=\frac{4}{x}\).

Step2: Solve for \(x\)

Cross - multiply: \(x\times\sqrt{3}=2\times4\). Then \(x=\frac{8}{\sqrt{3}}\).
Rationalize the denominator: \(x=\frac{8\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=\frac{8\sqrt{3}}{3}\).

Answer:

\(x = \frac{8\sqrt{3}}{3}\)