To solve the problem step-by-step, let’s break it down into two parts: finding [tex]\( b \)[/tex] and identifying the correct equation to use to solve for [tex]\( a \)[/tex].
Step 1: Solving for [tex]\( b \)[/tex]
We know that:
[tex]\[ \cos(22.6^\circ) = \frac{b}{13} \][/tex]
To isolate [tex]\( b \)[/tex], we multiply both sides of the equation by 13:
[tex]\[ b = 13 \cos(22.6^\circ) \][/tex]
Using a calculator, we find:
[tex]\[ \cos(22.6^\circ) \approx 0.9239 \][/tex]
Therefore:
[tex]\[ b = 13 \times 0.9239 \approx 12.0117 \][/tex]
Rounding [tex]\( 12.0117 \)[/tex] to the nearest whole number:
[tex]\[ b \approx 12 \][/tex]
Step 2: Identifying the correct equation to use for [tex]\( a \)[/tex]
We are provided with the following options for 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]
To determine which equation is correct, let's recall the definitions and relationships of trigonometric functions in a right triangle. Specifically:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
In triangle [tex]\( ABC \)[/tex], with [tex]\( \angle B = 22.6^\circ \)[/tex]:
- The side opposite [tex]\( \angle B \)[/tex] is [tex]\( a \)[/tex].
- The side adjacent to [tex]\( \angle B \)[/tex] is [tex]\( b \approx 12 \)[/tex].
From trigonometric properties:
[tex]\[ \tan(22.6^\circ) = \frac{a}{\text{adjacent side}} \][/tex]
Given the choices and knowing that [tex]\( b \approx 12 \)[/tex]:
[tex]\[ \tan(22.6^\circ) = \frac{a}{12} \][/tex]
Thus, the correct equation is:
[tex]\[ \tan(22.6^\circ) = \frac{a}{12} \][/tex]
Therefore, the answer is:
[tex]\[ \boxed{\tan(22.6^\circ) = \frac{a}{12}} \][/tex]
