To find the length of [tex]\(\overline{AB}\)[/tex] using the given equation [tex]\(\sin(25^{\circ}) = \frac{9}{c}\)[/tex], we need to follow these steps:
1. Understand the meaning of the equation:
- The equation [tex]\(\sin(25^{\circ}) = \frac{9}{c}\)[/tex] indicates a relationship between the angle, the opposite side (which is 9 units), and the hypotenuse (denoted by [tex]\(c\)[/tex]) of a right triangle.
2. Isolate the variable [tex]\(c\)[/tex]:
- To find [tex]\(c\)[/tex], we rearrange the equation to solve for [tex]\(c\)[/tex]:
[tex]\[
c = \frac{9}{\sin(25^{\circ})}
\][/tex]
3. Determine [tex]\(\sin(25^{\circ})\)[/tex]:
- You can use the sine function to determine the value of [tex]\(\sin(25^{\circ})\)[/tex]. This is a task normally performed with a calculator where you get a value of approximately [tex]\(\sin(25^{\circ}) \approx 0.4226\)[/tex].
4. Calculate [tex]\(c\)[/tex]:
- Plug the value of [tex]\(\sin(25^{\circ})\)[/tex] into the equation:
[tex]\[
c = \frac{9}{0.4226} \approx 21.295814248372487
\][/tex]
5. Round to the nearest tenth:
- Rounding 21.295814248372487 to the nearest tenth gives us 21.3.
Thus, the length of [tex]\(\overline{AB}\)[/tex] is [tex]\(21.3\)[/tex] inches. Therefore, the correct answer is:
21.3 in.
