The equation [tex]\sin \left(25^{\circ}\right)=\frac{9}{6}[/tex] can be used to find the length of [tex]\overline{AB}[/tex].

A. 19.3 in.
B. 21.3 in.
C. 23.5 in.
D. 68.0 in.



Answer :

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.