Sure, let's solve this question step-by-step to determine which length corresponds to [tex]\(\overline{AB}\)[/tex].
1. Understanding the Given Equation:
The given equation is:
[tex]\[
\sin(25^\circ) = \frac{9}{6}
\][/tex]
2. Identifying the Sine Relationship:
In the context of trigonometry, for a right-angled triangle:
[tex]\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\][/tex]
For [tex]\(\theta = 25^\circ\)[/tex]:
[tex]\[
\sin(25^\circ) = \frac{9}{6} = 1.5
\][/tex]
3. Checking the Validity of [tex]\(\sin(25^\circ) = 1.5\)[/tex]:
The sine of an angle cannot be greater than 1 because the range for sine values is [tex]\([-1, 1]\)[/tex]. Therefore, [tex]\( \sin(25^\circ) = 1.5\)[/tex] is not mathematically possible. This indicates that there might be a different use of the given values.
4. Reassessing the Calculation with Actual [tex]\(\sin(25^\circ)\)[/tex]:
Let's use the correct value of [tex]\(\sin(25^\circ)\)[/tex]:
[tex]\[
\sin(25^\circ) ≈ 0.4226
\][/tex]
5. Finding the Hypotenuse:
The provided information suggests [tex]\(9\)[/tex] is the opposite side to the angle [tex]\(25^\circ\)[/tex]. Let's set up the correct equation to find the hypotenuse:
[tex]\[
\sin(25^\circ) = \frac{\text{opposite}}{\text{hypotenuse}}
\][/tex]
So,
[tex]\[
\text{hypotenuse} = \frac{\text{opposite}}{\sin(25^\circ)} = \frac{9}{0.4226} \approx 21.3
\][/tex]
Hence, the length of [tex]\(\overline{AB}\)[/tex] is approximately [tex]\(21.3\)[/tex] inches.
This result matches the second option provided, which is [tex]\(21.3\)[/tex] inches.
