Question

find the value of the side a in the triangle below. round to two decimal places if necessary. the angle t, noted in blue below, measures 30°. draw

Explanation:

Step1: Identify the trigonometric ratio

The triangle is a right - triangle. We know that for an acute angle \(t = 30^{\circ}\) in a right - triangle, \(\tan(t)=\frac{\text{opposite}}{\text{adjacent}}\). The side opposite to the \(30^{\circ}\) angle is \(11\) and the side adjacent to the \(30^{\circ}\) angle is \(a\) (wait, no, wait. Wait, the right - angle is at the bottom left, the vertical side is \(a\), the horizontal side is \(11\), and the angle at the top is \(30^{\circ}\). So \(\tan(30^{\circ})=\frac{11}{a}\) (because the opposite side to \(30^{\circ}\) is \(11\) and the adjacent side is \(a\)).

Step2: Solve for \(a\)

We know that \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\approx0.5774\). From \(\tan(30^{\circ})=\frac{11}{a}\), we can re - arrange the formula to get \(a=\frac{11}{\tan(30^{\circ})}\).

Substitute \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\) into the formula: \(a = 11\sqrt{3}\).

Calculate the numerical value: \(11\sqrt{3}\approx11\times1.732 = 19.052\approx19.05\) (if we consider the other way: wait, maybe I mixed up opposite and adjacent. Wait, the angle is at the top, so the vertical side is adjacent, the horizontal side is opposite. So \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\), so \(\tan(30^{\circ})=\frac{11}{a}\), so \(a=\frac{11}{\tan(30^{\circ})}\). Since \(\tan(30^{\circ})=\frac{1}{\sqrt{3}}\), then \(a = 11\sqrt{3}\approx19.05\).

Wait, another way: if we use \(\cot(30^{\circ})=\sqrt{3}=\frac{\text{adjacent}}{\text{opposite}}\), so \(\cot(30^{\circ})=\frac{a}{11}\), so \(a = 11\cot(30^{\circ})\). Since \(\cot(30^{\circ})=\sqrt{3}\), then \(a = 11\sqrt{3}\approx19.05\).

Answer:

\(19.05\) (or \(11\sqrt{3}\approx19.05\))