Given a right triangle with angle $gfe$, where the side adjacent to angle $gfe$ has length $a$ and the hypotenuse has length $h$, which equation can be used to find the measure of angle $gfe$?

• A. $gfe={\cos }^{-1}\frac{h}{a}$
• B. $gfe={\sin }^{-1}\frac{a}{h}$
• C. $gfe={\sin }^{-1}\frac{h}{a}$
• D. $gfe={\cos }^{-1}\frac{a}{h}$

Answer: (D)

Verified step by step guidance

1. Identify the sides of the right triangle relative to the angle \( gfe \): the side adjacent to \( gfe \) is given as \( a \), and the hypotenuse is \( h \).
2. Recall the definition of the cosine function in a right triangle: \( \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \). Here, \( \theta = gfe \), so \( \cos(gfe) = \frac{a}{h} \).
3. To find the measure of the angle \( gfe \), use the inverse cosine function (also called arccosine), which gives the angle when the cosine value is known: \( gfe = \cos^{-1}\left( \frac{a}{h} \right) \).
4. Check the other options: using sine inverse would involve the opposite side over hypotenuse, which is not given here, so those options are incorrect.
5. Therefore, the correct equation to find the angle \( gfe \) is \( gfe = \cos^{-1}\left( \frac{a}{h} \right) \).