Question

which is the best approximation for the measure of angle egf?
32.8°
40.2°
49.8°
57.2°
(diagram: right triangle efg with right angle at e, ef = 12 in, fg = 18.6 in)

Explanation:

Step1: Identify triangle type and trigonometric ratio

Triangle \( EFG \) is right - angled at \( E \). We know the length of the opposite side to angle \( G \) (\( EF = 12\) in) and the hypotenuse (\( FG=18.6\) in). The sine ratio is given by \( \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} \), where \( \theta=\angle EGF \), opposite \( = EF = 12\), hypotenuse \( = FG = 18.6 \). So \( \sin(x)=\frac{12}{18.6} \).

Step2: Calculate the value of \( \sin(x) \)

\( \sin(x)=\frac{12}{18.6}\approx0.6452 \)

Step3: Find the angle \( x \)

To find \( x \), we take the inverse sine (arcsin) of \( 0.6452 \). So \( x=\arcsin(0.6452) \). Using a calculator, \( \arcsin(0.6452)\approx40.2^{\circ} \)

Answer:

\( 40.2^{\circ} \) (corresponding to the option with \( 40.2^{\circ} \))