Question

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

Explanation:

Step1: Identify the trigonometric ratio

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, for angle \(x\), the side opposite to \(x\) is \(GH = 8.2\) and the hypotenuse is \(HP\).

Step2: Apply the sine formula

\(\sin x=\frac{GH}{HP}\). Substituting \(GH = 8.2\) and \(HP=8.2 + 13\) (wait, no, in a right - triangle \(GHP\) with \(\angle G = 90^{\circ}\), by Pythagoras \(HP=\sqrt{8.2^{2}+13^{2}}=\sqrt{67.24 + 169}=\sqrt{236.24}\approx15.4\) (but actually, using the sine function directly: \(\sin x=\frac{GH}{HP}\), \(GH = 8.2\), \(GP = 13\), hypotenuse \(HP=\sqrt{8.2^{2}+13^{2}}\approx15.4\) (no, wait, in a right - triangle \(\sin x=\frac{\text{opposite}}{\text{hypotenuse}}\), the side opposite to \(x\) is \(GH = 8.2\), adjacent is \(GP = 13\), hypotenuse \(HP=\sqrt{8.2^{2}+13^{2}}\approx15.4\). But using \(\tan x=\frac{GH}{GP}\) (since \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\) in a right - triangle). So \(\tan x=\frac{8.2}{13}\).

Step3: Calculate \(x\)

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

Answer:

\(x\approx32.3^{\circ}\)