Question

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

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(g^{2}=e^{2}+f^{2}-2ef\cos G\). In \(\triangle EFG\), let \(e = 17\), \(f = 12\), \(g = 8\). We want to find \(\angle E\). The Law of Cosines formula for \(\angle E\) is \(EG^{2}=EF^{2}+FG^{2}-2(EF)(FG)\cos E\). Substituting the values: \(12^{2}=8^{2}+17^{2}-2\times8\times17\times\cos E\).

$$144 = 64 + 289-272\cos E$$

$$144=353 - 272\cos E$$

Step2: Solve for \(\cos E\)

Rearrange the equation:

$$272\cos E=353 - 144$$

$$272\cos E = 209$$

\(\cos E=\frac{209}{272}\approx0.768\)

Step3: Find the angle \(E\)

Take the inverse - cosine of \(0.768\). \(E=\cos^{-1}(0.768)\)
Using a calculator, \(E\approx40.0^{\circ}\)

Answer:

\(40.0\)