To determine the lengths of the legs of a right triangle given an acute angle of 19° and a hypotenuse of 15 units, we can follow these steps:
1. Identify the components:
- Hypotenuse ([tex]\(c\)[/tex]): 15 units
- Angle ([tex]\(\theta\)[/tex]): 19°
2. Use trigonometric functions to find the lengths of the legs:
- The leg opposite to the angle ([tex]\(a\)[/tex]) can be found using the sine function.
- The leg adjacent to the angle ([tex]\(b\)[/tex]) can be found using the cosine function.
3. Calculate the length of the leg opposite the angle using the sine function:
[tex]\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\][/tex]
So, the length of the opposite leg is:
[tex]\[
\text{opposite} = \sin(19°) \times 15
\][/tex]
4. Calculate the length of the leg adjacent to the angle using the cosine function:
[tex]\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\][/tex]
So, the length of the adjacent leg is:
[tex]\[
\text{adjacent} = \cos(19°) \times 15
\][/tex]
5. Round the results to the nearest tenth for both values.
Following these steps, we find that:
- The length of the leg opposite the 19° angle is approximately [tex]\(4.9\)[/tex] units.
- The length of the leg adjacent to the 19° angle is approximately [tex]\(14.2\)[/tex] units.
Therefore, the correct answer is:
D. [tex]\(4.9\)[/tex] units, [tex]\(14.2\)[/tex] units
