Given a right triangle with angle $\mathrm{FGE}$ and the side opposite to $\mathrm{FGE}$ has length $a$, the side adjacent to $\mathrm{FGE}$ has length $b$, and the hypotenuse has length $c$, which equation can be used to find the measure of angle $\mathrm{FGE}$?

• A. $\cos \left(\mathrm{FGE}\right)=\frac{c}{a}$
• B. $\sin \left(\mathrm{FGE}\right)=\frac{b}{c}$
• C. $\cos \left(\mathrm{FGE}\right)=\frac{a}{b}$
• D. $\sin \left(\mathrm{FGE}\right)=\frac{a}{c}$

Answer: (D)

Verified step by step guidance

1. Identify the sides relative to angle FGE in the right triangle: the side opposite to angle FGE is of length \(a\), the side adjacent to angle FGE is of length \(b\), and the hypotenuse is of length \(c\).
2. Recall the definitions of the primary trigonometric ratios for a right triangle: \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\), and \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\).
3. Since we want to find an equation involving angle FGE and the side lengths, match the correct ratio to the given sides: the sine of angle FGE is the ratio of the opposite side \(a\) to the hypotenuse \(c\), so \(\sin(\text{FGE}) = \frac{a}{c}\).
4. Check the other given options to see if they correctly represent any trigonometric ratio: for example, \(\cos(\text{FGE})\) should be \(\frac{b}{c}\), not \(\frac{c}{a}\) or \(\frac{a}{b}\), and \(\sin(\text{FGE})\) should not be \(\frac{b}{c}\).
5. Conclude that the correct equation to find the measure of angle FGE is \(\sin(\text{FGE}) = \frac{a}{c}\), which can be used to solve for angle FGE by taking the inverse sine (arcsin) of both sides.