Find the acute angle [tex]\theta[/tex] for the given trigonometric function. Round off the results according to the provided table.

[tex]\sin \theta = 0.8314[/tex]

\begin{tabular}{cc}
\hline
\begin{tabular}{c}
Measurements of \\
Angle to Nearest
\end{tabular} & \begin{tabular}{c}
Accuracy of \\
Trigonometric \\
Function
\end{tabular} \\
\hline
[tex]$1^{\circ}$[/tex] & 2 significant digits \\
[tex]$0.1^{\circ}$[/tex] & 3 significant digits \\
[tex]$0.01^{\circ}$[/tex] & 4 significant digits \\
\hline
\end{tabular}

Select the correct choice and, if necessary, fill in the answer box to complete your choice.

A. [tex]\theta = \square \square^{\circ}[/tex]
(Do not include the degree symbol in your answer.)

B. This value is not possible.



Answer :

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).