Question

find the value of x. you may round your answer to two decimal places.

Explanation:

Step1: Identify trigonometric ratio

In the right triangle, we have the opposite side (30) to the angle \(52^\circ\) and we need to find the adjacent side \(x\). The tangent function relates opposite and adjacent sides: \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\).

Step2: Set up the equation

\(\tan(52^\circ)=\frac{30}{x}\) (Wait, no: Wait, actually, \(x\) is adjacent? Wait, no, looking at the triangle: the right angle is at the top, so the angle at the bottom is \(52^\circ\), the side opposite to \(52^\circ\) is \(x\), and the adjacent side is 30? Wait, no, let's re - check. The right angle is between \(x\) and 30. So the angle at the bottom is \(52^\circ\), so the side opposite to \(52^\circ\) is \(x\), and the adjacent side is 30. Wait, no, \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\), so \(\tan(52^\circ)=\frac{x}{30}\)? Wait, no, if the angle is \(52^\circ\), the side adjacent to \(52^\circ\) is 30, and the side opposite is \(x\). So \(\tan(52^\circ)=\frac{x}{30}\), then \(x = 30\times\tan(52^\circ)\).

Step3: Calculate the value

We know that \(\tan(52^\circ)\approx1.2799\). So \(x = 30\times1.2799\approx38.397\approx38.40\) (Wait, no, wait, maybe I mixed up opposite and adjacent. Wait, let's look again. The right angle is at the top, so the two legs are \(x\) (horizontal) and 30 (vertical). The angle at the bottom is \(52^\circ\), so the angle between the vertical leg (30) and the hypotenuse is \(52^\circ\). So the horizontal leg \(x\) is opposite to \(52^\circ\), and the vertical leg 30 is adjacent to \(52^\circ\). So \(\tan(52^\circ)=\frac{x}{30}\), so \(x = 30\times\tan(52^\circ)\).
Calculating \(\tan(52^\circ)\approx1.2799\), then \(x = 30\times1.2799 = 38.397\approx38.40\). Wait, but maybe I had it reversed. Wait, if the angle is \(52^\circ\), and the vertical side is 30, and we want \(x\) (horizontal), then \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\), opposite is \(x\), adjacent is 30, so \(\tan(52)=\frac{x}{30}\), so \(x = 30\tan(52)\).
\(\tan(52^\circ)\approx1.2799\), so \(x = 30\times1.2799 = 38.397\approx38.40\).

Answer:

\(x\approx38.40\)