Let's solve the problem step-by-step.
### Step 1: Solving for [tex]\(D\)[/tex]
Given the equation:
[tex]\[
\cos \left(22.6^{\circ}\right) = \frac{D}{13}
\][/tex]
We need to solve for [tex]\(D\)[/tex]. Rearranging the equation for [tex]\(D\)[/tex]:
[tex]\[
D = 13 \times \cos \left(22.6^{\circ}\right)
\][/tex]
Using the given calculations, we find:
[tex]\[
D = 12
\][/tex]
### Step 2: Identifying the Correct Equation to Solve for [tex]\(b\)[/tex]
Given a new set of options to identify which equation correctly uses [tex]\(b\)[/tex] (which is the value we just calculated as [tex]\(D\)[/tex]):
[tex]\[
\tan \left(22.6^{\circ}\right)=\frac{a}{13}
\][/tex]
[tex]\[
\tan \left(22.6^{\circ}\right)=\frac{13}{a}
\][/tex]
[tex]\[
\tan \left(22.6^{\circ}\right)=\frac{a}{12}
\][/tex]
[tex]\[
\tan \left(22.6^{\circ}\right)=\frac{12}{a}
\][/tex]
To determine which equation is correct, let's recall the relationship between the sides in a right triangle involving [tex]\(\tan\)[/tex]:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
In this scenario:
- [tex]\(\theta = 22.6^{\circ}\)[/tex]
- The side adjacent to [tex]\(\theta\)[/tex] is the one where we have used cosine, which corresponds to side [tex]\(D = 12\)[/tex].
- The side opposite to [tex]\(\theta\)[/tex] is [tex]\(a\)[/tex].
Thus, the correct equation using the tangent function would be:
[tex]\[
\tan \left(22.6^{\circ}\right)=\frac{a}{12}
\][/tex]
### Summary of the Solution
1. [tex]\(D\)[/tex] was found to be [tex]\(12\)[/tex].
2. The correct equation using the value of [tex]\(b\)[/tex] to solve for [tex]\(a\)[/tex] is:
[tex]\[
\tan \left(22.6^{\circ}\right)=\frac{a}{12}
\][/tex]
Thus, the correct final output is:
[tex]\[
(12, 'The correct equation is tan(22.6°) = a / 12')
\][/tex]
