Question

find ( mangle h ).
write your answer as an integer or as a decimal rounded to the nearest tenth.
( mangle h=) (square^{circ})

Explanation:

Step1: Use trigonometric ratio

In right - triangle \(FGH\) with \(\angle G = 90^{\circ}\), we use the tangent function. The tangent of an angle in a right - triangle is defined as \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). But here, we can also use the sine function. The sine of an angle \(\angle H\) is given by \(\sin H=\frac{FG}{FH}\). First, find the length of \(FH\) using the Pythagorean theorem \(FH=\sqrt{FG^{2}+GH^{2}}\). Given \(FG = 4\) and \(GH=5\), then \(FH=\sqrt{4^{2}+5^{2}}=\sqrt{16 + 25}=\sqrt{41}\).

Another way: Using the tangent function. \(\tan H=\frac{FG}{GH}\), but more straightforward is using the inverse trigonometric function. Since \(\sin H=\frac{FG}{FH}\) (where \(FH=\sqrt{41}\)), also, using the fact that \(\tan H=\frac{FG}{GH}\), \(\tan H=\frac{4}{5}\).

Step2: Calculate the angle

We know that if \(\tan\theta=\frac{4}{5}\), then \(\theta=\tan^{- 1}(\frac{4}{5})\). Using a calculator, \(\tan^{-1}(x)\) where \(x = 0.8\). \(\tan^{-1}(0.8)\approx38.7^{\circ}\)

Answer:

\(38.7\)