To determine the measure of the unknown acute angle in the right triangle, we follow these steps:
1. Understand the Problem:
We are given:
- The length of the opposite side to the unknown angle is 4.8 feet.
- The length of the hypotenuse is 5.0 feet.
We need to find the measure of the angle.
2. Trigonometric Function:
We will use the sine function, which is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.
[tex]\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\][/tex]
Substituting the given values:
[tex]\[
\sin(\theta) = \frac{4.8}{5.0}
\][/tex]
3. Calculate the Sine Value:
Simplify the fraction:
[tex]\[
\sin(\theta) = 0.96
\][/tex]
4. Determine the Angle:
To find the angle [tex]\( \theta \)[/tex], we need to use the inverse sine function (also known as arcsine), which will give us the angle whose sine is 0.96.
[tex]\[
\theta = \sin^{-1}(0.96)
\][/tex]
5. Convert to Degrees:
After calculating the angle [tex]\( \theta \)[/tex] in radians, we convert it to degrees.
The result is:
[tex]\[
\theta \approx 73.7^{\circ}
\][/tex]
Thus, the measure of the unknown angle is approximately [tex]\( 73.7^{\circ} \)[/tex].
Therefore, the correct answer is:
[tex]\[
73.7^{\circ}
\][/tex]
