Question

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

Explanation:

Step1: Use the Law of Cosines

The Law of Cosines formula is \(e^{2}=f^{2}+g^{2}-2fg\cos E\). For \(\triangle EFG\), to find \(\angle F\), we use the formula \(\cos F=\frac{EF^{2}+FG^{2}-EG^{2}}{2\cdot EF\cdot FG}\). Here, \(EF = 14\), \(FG=22\), \(EG = 10\).
Substitute the values: \(\cos F=\frac{14^{2}+22^{2}-10^{2}}{2\times14\times22}\).
First, calculate the numerator: \(14^{2}=196\), \(22^{2}=484\), \(10^{2} = 100\). Then \(196 + 484-100=580\).
The denominator is \(2\times14\times22=616\). So \(\cos F=\frac{580}{616}\approx0.94156\).

Step2: Find the angle

Take the inverse - cosine of \(0.94156\). \(F=\cos^{-1}(0.94156)\).
Using a calculator, \(F\approx19.7^{\circ}\).

Answer:

\(19.7\)