Question

16
in △def, if side ( d = 14 ), side ( e = 10 ), and side ( f = 12 ), find ( mangle f ) to the nearest degree.
answer

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(f^{2}=d^{2}+e^{2}-2de\cos F\).
Substitute \(d = 14\), \(e = 10\), and \(f = 12\) into the formula:
\(12^{2}=14^{2}+10^{2}-2\times14\times10\times\cos F\).

Step2: Simplify the equation

First, calculate the squares:
\(144=196 + 100-280\cos F\).
Then, combine like - terms:
\(144=296-280\cos F\).

Step3: Solve for \(\cos F\)

Rearrange the equation to isolate \(\cos F\):
\(280\cos F=296 - 144\).
\(280\cos F=152\).
\(\cos F=\frac{152}{280}\approx0.5429\).

Step4: Find the angle \(F\)

Take the inverse cosine (\(\cos^{-1}\)) of \(0.5429\):
\(F=\cos^{-1}(0.5429)\approx57^{\circ}\).

Answer:

\(57^{\circ}\)