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: Use the Law of Cosines

The Law of Cosines formula is \(d^{2}=e^{2}+f^{2}-2ef\cos D\). For \(\triangle DEF\) with \(d = DF = 10\), \(e=EF = 8\), \(f = DE=9\), and we want to find \(\angle E\). The formula for \(\cos E\) is \(\cos E=\frac{DE^{2}+EF^{2}-DF^{2}}{2\cdot DE\cdot EF}\).
Substitute the values: \(DE = 9\), \(EF = 8\), \(DF = 10\) into the formula.
\(\cos E=\frac{9^{2}+8^{2}-10^{2}}{2\times9\times8}\)

Step2: Calculate the numerator and denominator

First, calculate the numerator: \(9^{2}+8^{2}-10^{2}=81 + 64-100=45\)
Then, calculate the denominator: \(2\times9\times8 = 144\)
So, \(\cos E=\frac{45}{144}=\frac{5}{16}\approx0.3125\)

Step3: Find the angle \(E\)

Since \(\cos E\approx0.3125\), then \(E=\cos^{-1}(0.3125)\)
Using a calculator, \(E\approx71.8^{\circ}\)

Answer:

\(71.8\)