Question

in △def, de = 18, ef = 16, and df = 12. find m∠f.
m∠f = type your answer_

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula for finding an angle in a triangle $\triangle DEF$ is $\cos F=\frac{DF^{2}+EF^{2}-DE^{2}}{2\cdot DF\cdot EF}$.
Substitute $DE = 18$, $EF = 16$, and $DF = 12$ into the formula:
$\cos F=\frac{12^{2}+16^{2}-18^{2}}{2\times12\times16}$.

Step2: Calculate the numerator and denominator

First, calculate the numerator:
$12^{2}=144$, $16^{2}=256$, $18^{2}=324$.
$12^{2}+16^{2}-18^{2}=144 + 256-324=76$.
Then, calculate the denominator:
$2\times12\times16 = 384$.
So, $\cos F=\frac{76}{384}=\frac{19}{96}$.

Step3: Find the angle measure

$F=\cos^{-1}(\frac{19}{96})$.
Using a calculator, $F\approx78.6^{\circ}$.

Answer:

$78.6^{\circ}$