The side opposite an acute angle of a right triangle measures 4.8 feet, and the hypotenuse measures 5.0 feet. What is the measure of the unknown angle?

A. [tex]\(73.7^\circ\)[/tex]
B. [tex]\(46.2^\circ\)[/tex]
C. [tex]\(43.8^\circ\)[/tex]
D. [tex]\(16.3^\circ\)[/tex]



Answer :

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]