To find the acute angle [tex]\(\theta\)[/tex] given that [tex]\(\csc \theta = 1.516\)[/tex], we need to follow these steps:
1. Understanding [tex]\(\csc \theta\)[/tex]: The cosecant function is the reciprocal of the sine function, i.e., [tex]\(\csc \theta = \frac{1}{\sin \theta}\)[/tex].
2. Calculating [tex]\(\sin \theta\)[/tex]:
We have [tex]\(\csc \theta = 1.516\)[/tex].
Therefore, [tex]\(\sin \theta = \frac{1}{\csc \theta} = \frac{1}{1.516} \approx 0.6594\)[/tex].
3. Finding [tex]\(\theta\)[/tex]:
To find [tex]\(\theta\)[/tex], we take the inverse sine (arcsin) of [tex]\(0.6594\)[/tex]:
[tex]\(\theta = \arcsin(0.6594) \approx 41.2717\)[/tex] degrees.
4. Rounding to the nearest [tex]\(0.1^{\circ}\)[/tex]:
According to the problem, we need to round our final answer to the nearest [tex]\(0.1^{\circ}\)[/tex].
Therefore, [tex]\(\theta \approx 41.3^{\circ}\)[/tex].
5. Determining the correct choice:
According to the given options and the required rounding accuracy (3 significant digits when rounding to [tex]\(0.1^{\circ}\)[/tex]), we check if this rounded value falls within an acute angle range. An acute angle is less than [tex]\(90^{\circ}\)[/tex].
The calculated angle [tex]\(\theta\)[/tex] is approximately [tex]\(41.3^{\circ}\)[/tex], and [tex]\(41.3\)[/tex] rounded to the nearest whole number compatible with the given choices is [tex]\(41.0\)[/tex].
Hence, the choice is:
A. [tex]\(\theta = 41.0\)[/tex]
So the final answer is:
[tex]\[
\theta = 41.0
\][/tex]
