Let's solve the problem step-by-step.
### Step 1: Use the Cosine Value to Find [tex]\( b \)[/tex]
We know that:
[tex]\[ \cos(22.6^\circ) = \frac{b}{13} \][/tex]
Rearrange the equation to solve for [tex]\( b \)[/tex]:
[tex]\[ b = 13 \cdot \cos(22.6^\circ) \][/tex]
Using the given value for [tex]\(\cos(22.6^\circ)\)[/tex], we can determine [tex]\( b \)[/tex]:
[tex]\[ b \approx 13 \cdot 0.927 \][/tex]
[tex]\[ b \approx 12.001732822468751 \][/tex]
### Step 2: Round [tex]\( b \)[/tex] to the Nearest Whole Number
Rounding [tex]\( b \)[/tex] to the nearest whole number, we get:
[tex]\[ b \approx 12 \][/tex]
### Step 3: Determine the Correct Equation to Solve for [tex]\( a \)[/tex]
Given the choices for the equations involving [tex]\(\tan(22.6^\circ)\)[/tex]:
1. [tex]\(\tan(22.6^\circ) = \frac{a}{13}\)[/tex]
2. [tex]\(\tan(22.6^\circ) = \frac{13}{a}\)[/tex]
3. [tex]\(\tan(22.6^\circ) = \frac{a}{12}\)[/tex]
4. [tex]\(\tan(22.6^\circ) = \frac{12}{a}\)[/tex]
We need to check which equation correctly relates [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\tan(22.6^\circ)\)[/tex].
Using the identity for a right triangle:
[tex]\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
In this case, rethink the sides of the triangle:
- [tex]\(\theta = 22.6^\circ\)[/tex]
- [tex]\( \text{adjacent side} = b \approx 12 \)[/tex]
- Hypotenuse = 13
Since:
[tex]\[ a = \text{opposite} \][/tex]
[tex]\[ b = \text{adjacent} \approx 12 \][/tex]
So:
[tex]\[ \tan(22.6^\circ) = \frac{a}{12} \][/tex]
Let's validate:
The equation that correctly uses [tex]\( b \approx 12 \)[/tex] to solve for [tex]\( a \)[/tex] is:
[tex]\[ \tan(22.6^\circ) = \frac{a}{12} \][/tex]
### Conclusion
The correct equation that uses the value of [tex]\( b \approx 12 \)[/tex] to solve for [tex]\( a \)[/tex] is:
[tex]\[ \tan(22.6^\circ) = \frac{a}{12} \][/tex]
Thus, the correct answer is:
[tex]\[ \tan(22.6^\circ) = \frac{a}{12} \][/tex]
