To find the acute angle [tex]\( \theta \)[/tex] for which [tex]\( \sin \theta = 0.8314 \)[/tex], we can follow these steps:
1. Identify the given trigonometric value:
[tex]\[ \sin \theta = 0.8314 \][/tex]
2. Use the inverse sine function to determine the angle [tex]\( \theta \)[/tex]:
- The inverse sine function, denoted as [tex]\( \sin^{-1} \)[/tex] or [tex]\( \arcsin \)[/tex], provides the angle whose sine is the given value.
3. Convert the angle from radians to degrees:
- The output of the [tex]\( \arcsin \)[/tex] function is typically in radians. To convert radians to degrees, we multiply by [tex]\( \frac{180}{\pi} \)[/tex].
4. Round the result:
- According to the provided table, we need to round the resulting angle to 2 significant digits since angles are measured to the nearest [tex]\( 1^\circ \)[/tex].
Considering these steps, the calculated value of the angle [tex]\( \theta \)[/tex] for which [tex]\( \sin \theta = 0.8314 \)[/tex] is approximately [tex]\( \theta = 56.24^\circ \)[/tex] before rounding.
Since we are required to round to two significant digits, the rounded value of the angle is:
[tex]\[
\theta = 56.24 \approx 56.24
\][/tex]
Thus, we select option A and fill in the answer box with [tex]\( \theta = 56.24 \)[/tex]:
[tex]\[
\boxed{56.24}
\][/tex]
So, the acute angle [tex]\( \theta \)[/tex] that satisfies [tex]\( \sin \theta = 0.8314 \)[/tex], rounded according to the provided table, is [tex]\( 56.24^\circ \)[/tex] (without including the degree symbol in the answer box).
