To solve the problem of finding the length of [tex]\(\overline{AB}\)[/tex], we can use the given equation [tex]\(\sin(25^\circ) = \frac{9}{c}\)[/tex], where [tex]\(c\)[/tex] represents the length of [tex]\(\overline{AB}\)[/tex]. Here's a step-by-step solution:
1. Identify the known values and the trigonometric function to be used:
- The angle given is [tex]\(25^\circ\)[/tex].
- The length of the opposite side to the angle [tex]\(25^\circ\)[/tex] is 9 units.
- We need to find the length of the hypotenuse [tex]\(c\)[/tex].
2. Write the trigonometric equation involving the sine function:
[tex]\[
\sin(25^\circ) = \frac{9}{c}
\][/tex]
3. Rearrange the equation to solve for [tex]\(c\)[/tex]:
[tex]\[
c = \frac{9}{\sin(25^\circ)}
\][/tex]
4. Calculate the sine of [tex]\(25^\circ\)[/tex]:
- Use a calculator to find [tex]\(\sin(25^\circ)\)[/tex]. The approximate value is:
[tex]\[
\sin(25^\circ) \approx 0.4226
\][/tex]
5. Substitute the value of [tex]\(\sin(25^\circ)\)[/tex] into the equation:
[tex]\[
c = \frac{9}{0.4226}
\][/tex]
6. Perform the division to find the length of the hypotenuse:
[tex]\[
c \approx 21.2958
\][/tex]
7. Match this calculated value with the given choices:
- The choices are: 19.3, 21.3, 23.5, and 68.0.
- The closest value to [tex]\(21.2958\)[/tex] is [tex]\(21.3\)[/tex].
8. Conclusion:
The length of [tex]\(\overline{AB}\)[/tex] is closest to [tex]\(21.3\)[/tex] inches.
Therefore, the length of [tex]\(\overline{AB}\)[/tex] is approximately [tex]\(21.3\)[/tex] inches.
