Question

solve for x
image of a right triangle with right angle at the bottom left, vertical leg labeled x, horizontal leg labeled 8, and the angle at the top left (between vertical leg and hypotenuse) labeled 30°

Explanation:

Step1: Identify trigonometric ratio

This is a right - triangle. We know one angle (\(30^{\circ}\)), the opposite side to the angle is \(8\) and the adjacent side is \(x\). The tangent function is defined as \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). So \(\tan(30^{\circ})=\frac{8}{x}\).

Step2: Solve for \(x\)

We know that \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\approx\frac{\sqrt{3}}{3}\). From \(\tan(30^{\circ})=\frac{8}{x}\), we can cross - multiply to get \(x\times\tan(30^{\circ}) = 8\). Then \(x=\frac{8}{\tan(30^{\circ})}\). Substituting \(\tan(30^{\circ})=\frac{\sqrt{3}}{3}\), we have \(x = \frac{8}{\frac{\sqrt{3}}{3}}=8\times\frac{3}{\sqrt{3}}=\frac{24}{\sqrt{3}}\). Rationalizing the denominator (multiplying numerator and denominator by \(\sqrt{3}\)), we get \(x=\frac{24\sqrt{3}}{3} = 8\sqrt{3}\approx13.86\).

Answer:

\(x = 8\sqrt{3}\) (or approximately \(13.86\))