To find the length of [tex]\(\overline{AB}\)[/tex] given the equation [tex]\(\sin(25^\circ) = \frac{9}{\overline{AB}}\)[/tex], we need to follow these steps:
1. Understand the Trigonometric Relationship:
The equation [tex]\(\sin(25^\circ) = \frac{9}{\overline{AB}}\)[/tex] can be interpreted within the context of a right triangle, where:
- 25° is one of the angles.
- The length of the side opposite the 25° angle is 9 units.
- [tex]\(\overline{AB}\)[/tex] is the hypotenuse of the triangle.
2. Isolate [tex]\(\overline{AB}\)[/tex]:
We start with the equation:
[tex]\[
\sin(25^\circ) = \frac{9}{\overline{AB}}
\][/tex]
To solve for [tex]\(\overline{AB}\)[/tex], we multiply both sides by [tex]\(\overline{AB}\)[/tex] and then divide by [tex]\(\sin(25^\circ)\)[/tex]:
[tex]\[
\overline{AB} \sin(25^\circ) = 9
\][/tex]
[tex]\[
\overline{AB} = \frac{9}{\sin(25^\circ)}
\][/tex]
3. Calculate [tex]\(\sin(25^\circ)\)[/tex]:
The sine of 25 degrees is a known trigonometric value which can be found using a calculator or trigonometric tables. In this case, we know from the result that:
[tex]\[
\sin(25^\circ) \approx 0.4226
\][/tex]
4. Solve for [tex]\(\overline{AB}\)[/tex]:
Plug in the value of [tex]\(\sin(25^\circ)\)[/tex]:
[tex]\[
\overline{AB} = \frac{9}{0.4226} \approx 21.295814248372487
\][/tex]
5. Round the Answer:
To get the final answer, we round 21.295814248372487 to the nearest tenth:
[tex]\[
\overline{AB} \approx 21.3
\][/tex]
Therefore, the length of [tex]\(\overline{AB}\)[/tex] is approximately 21.3 inches. The correct answer is:
[tex]\[
\boxed{21.3 \text{ in.}}
\][/tex]
