Question

1. find x.

image of a right triangle with a right angle at the bottom left, one leg labeled 11 (horizontal), the other leg labeled x (vertical), and an angle of 17° between the vertical leg and the hypotenuse

Explanation:

Step1: Identify the trigonometric ratio

We have a right triangle, and we know the angle \(17^\circ\), the opposite side to the angle is \(11\) (wait, no, wait: the side adjacent to the \(17^\circ\) angle is \(x\), and the opposite side is \(11\)? Wait, no, let's check the triangle. The right angle is at the bottom left, so the vertical side is \(x\) (adjacent to \(17^\circ\)), the horizontal side is \(11\) (opposite to \(17^\circ\)). So we use tangent: \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\). So \(\tan(17^\circ)=\frac{11}{x}\). Wait, no: \(\theta = 17^\circ\), opposite side is \(11\), adjacent side is \(x\). So \(\tan(17^\circ)=\frac{11}{x}\), so we can solve for \(x\): \(x=\frac{11}{\tan(17^\circ)}\).

Step2: Calculate the value

First, find \(\tan(17^\circ)\). Using a calculator, \(\tan(17^\circ)\approx0.3057\). Then \(x=\frac{11}{0.3057}\approx35.98\) (or more accurately, let's compute it properly).

Wait, wait, maybe I mixed up opposite and adjacent. Let's re-express: in the right triangle, the angle at the top (between the hypotenuse and the vertical side \(x\)) is \(17^\circ\). So the vertical side is adjacent to \(17^\circ\), the horizontal side (length 11) is opposite to \(17^\circ\). So \(\tan(17^\circ)=\frac{\text{opposite}}{\text{adjacent}}=\frac{11}{x}\). So solving for \(x\): \(x = \frac{11}{\tan(17^\circ)}\).

Calculating \(\tan(17^\circ)\): using calculator, \(\tan(17^\circ)\approx0.3057307787\). Then \(x=\frac{11}{0.3057307787}\approx35.98\), which is approximately 36.

Answer:

\(x\approx36\) (or more precisely, \(\frac{11}{\tan(17^\circ)}\approx35.98\))