Question

find ( mangle f ).
write your answer as an integer or as a decimal rounded to the nearest tenth.
( mangle f=) 。

Explanation:

Step1: Use the sine function

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For \(\angle F\), the side opposite to \(\angle F\) is \(GH = 8\) and the hypotenuse is \(GF\). First, find \(GF\) using the Pythagorean theorem \(GF=\sqrt{GH^{2}+FH^{2}}=\sqrt{8^{2} + 4^{2}}=\sqrt{64 + 16}=\sqrt{80}=4\sqrt{5}\). Then \(\sin F=\frac{GH}{GF}=\frac{8}{4\sqrt{5}}=\frac{2}{\sqrt{5}}\).

Step2: Calculate the angle

\(m\angle F=\sin^{- 1}(\frac{2}{\sqrt{5}})\). Using a calculator, \(\sin^{-1}(\frac{2}{\sqrt{5}})\approx63.4^{\circ}\)

Answer:

\(63.4\)