Question

solve for x. round to the nearest tenth of a degree, if necessary.

Explanation:

Step1: Use trigonometric ratio

In a right - triangle, the sine of an angle is defined as \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, for angle \(x\), the side opposite to \(x\) is \(GH = 8.2\) and the hypotenuse is \(HF\). But first, we need to confirm the trigonometric ratio. Wait, no, \(\sin x=\frac{GH}{HF}\) (opposite over hypotenuse in right - triangle \( \triangle GHF\) with \(\angle G = 90^{\circ}\)).

Step2: Substitute values

We know \(GH = 8.2\) and \(GF=13\). Using the sine function \(\sin x=\frac{GH}{HF}\) (wait, no, correct ratio: \(\sin x=\frac{GH}{HF}\) is wrong. Wait, in right - triangle \(\triangle GHF\), \(\sin x=\frac{GH}{HF}\) (opposite = \(GH\), hypotenuse=\(HF\)). But wait, no, \(\sin x=\frac{\text{opposite}}{\text{hypotenuse}}\), here opposite to \(x\) is \(GH = 8.2\), hypotenuse \(HF=\sqrt{8.2^{2}+13^{2}}\) (no, wait, no! Wait, in right - triangle \(\angle G = 90^{\circ}\), so \(\sin x=\frac{GH}{HF}\) (opposite = \(GH\), hypotenuse=\(HF\)). Wait, no, \(\sin x=\frac{GH}{HF}\) (opposite = \(GH\), hypotenuse=\(HF\)). Wait, no! Wait, \(\sin x=\frac{\text{opposite}}{\text{hypotenuse}}\), for angle \(x\), opposite side is \(GH = 8.2\), hypotenuse is \(HF\) (but we can also use \(\sin x=\frac{GH}{HF}\) (wait, no, \(\sin x=\frac{\text{opposite}}{\text{hypotenuse}}\), here \(\text{opposite}=GH = 8.2\), \(\text{hypotenuse}\) is the side opposite the right angle. Wait, in \(\triangle GHF\) with \(\angle G = 90^{\circ}\), hypotenuse is \(HF\). So \(\sin x=\frac{GH}{HF}\). But \(GH = 8.2\), \(GF = 13\) (adjacent to \(x\)). Wait, no! \(\tan x=\frac{GH}{GF}\) (opposite over adjacent).

So \(\tan x=\frac{GH}{GF}\) (since \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\) in a right - triangle). Substitute \(GH = 8.2\) and \(GF = 13\). Then \(x=\tan^{- 1}(\frac{8.2}{13})\).

Step3: Calculate the value

\(\frac{8.2}{13}\approx0.6308\). Then \(x=\tan^{-1}(0.6308)\). Using a calculator, \(x\approx32.3^{\circ}\).

Answer:

\(32.3^{\circ}\)