Question

which equation can be used to find the measure of angle gfe?
\\( \cos ^ { - 1 } \left( \frac { 14.5 } { 11.9 } \
ight) = \theta \\)
\\( \cos ^ { - 1 } \left( \frac { 11.9 } { 14.5 } \
ight) = \theta \\)
\\( \sin ^ { - 1 } \left( \frac { 11.9 } { 14.5 } \
ight) = \theta \\)
\\( \sin ^ { - 1 } \left( \frac { 14.5 } { 11.9 } \
ight) = \theta \\)

Explanation:

Step1: Recall the cosine formula

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\).

Step2: Identify the sides relative to \(\theta\)

In right - triangle \(GEF\) with right - angle at \(E\), the hypotenuse \(GF = 14.5\) and the adjacent side to \(\theta\) (angle \(GFE\)) is \(FE=11.9\).

Step3: Apply the inverse cosine function

Since \(\cos\theta=\frac{FE}{GF}\), then \(\theta=\cos^{- 1}(\frac{FE}{GF})\). Substituting \(FE = 11.9\) and \(GF = 14.5\), we get \(\theta=\cos^{-1}(\frac{11.9}{14.5})\).

Answer:

\(\cos^{-1}(\frac{11.9}{14.5})=\theta\) (the second option)