To solve the given problem, we start by addressing the cosine and tangent relationships within the right triangle.
Step 1: Solve for [tex]\( b \)[/tex] using [tex]\(\cos(22.6^\circ) = \frac{b}{13}\)[/tex]
Given:
[tex]\[
\cos(22.6^\circ) = \frac{b}{13}
\][/tex]
To find [tex]\( b \)[/tex]:
[tex]\[
b = 13 \cdot \cos(22.6^\circ)
\][/tex]
Given that the result is approximately:
[tex]\[
b \approx 12
\][/tex]
After rounding, [tex]\( b \)[/tex] to the nearest whole number is [tex]\( 12 \)[/tex].
Step 2: Identify the correct equation using the value of [tex]\( b \)[/tex] and the tangent function to solve for [tex]\( a \)[/tex]
We need to find which of the given equations involving [tex]\(\tan(22.6^\circ)\)[/tex] is correctly related to the sides of the triangle.
Equation Analysis:
1. [tex]\(\tan(22.6^\circ) = \frac{a}{13}\)[/tex]
Evaluate:
[tex]\[
a = 13 \cdot \tan(22.6^\circ) \approx 5.411
\][/tex]
2. [tex]\(\tan(22.6^\circ) = \frac{13}{a}\)[/tex]
Evaluate:
[tex]\[
a = \frac{13}{\tan(22.6^\circ)} \approx 31.230
\][/tex]
3. [tex]\(\tan(22.6^\circ) = \frac{a}{12}\)[/tex]
Evaluate:
[tex]\[
a = 12 \cdot \tan(22.6^\circ) \approx 4.995
\][/tex]
4. [tex]\(\tan(22.6^\circ) = \frac{12}{a}\)[/tex]
Evaluate:
[tex]\[
a = \frac{12}{\tan(22.6^\circ)} \approx 28.828
\][/tex]
These values are derived based on the tangent function of the given angle.
Conclusion:
By evaluating the tangent relationships:
- [tex]\(\tan(22.6^\circ) = \frac{a}{13}\)[/tex] gives [tex]\( a \approx 5.411 \)[/tex]
- [tex]\(\tan(22.6^\circ) = \frac{13}{a}\)[/tex] gives [tex]\( a \approx 31.230 \)[/tex]
- [tex]\(\tan(22.6^\circ) = \frac{a}{12}\)[/tex] gives [tex]\( a \approx 4.995 \)[/tex]
- [tex]\(\tan(22.6^\circ) = \frac{12}{a}\)[/tex] gives [tex]\( a \approx 28.828 \)[/tex]
The accurate solution based on these calculations corresponds to the relationships given and the values derived. Therefore, we verify which value makes sense for a right triangle with [tex]\( \cos \left(22.6^\circ\right)=\frac{b}{13} \)[/tex]:
The correct equations are:
- [tex]\(\tan(22.6^\circ) = \frac{a}{13}\)[/tex]
- [tex]\(\tan(22.6^\circ) = \frac{13}{a}\)[/tex]
- [tex]\(\tan(22.6^\circ) = \frac{a}{12}\)[/tex]
- [tex]\(\tan(22.6^\circ) = \frac{12}{a}\)[/tex]
Given the confirmed values from the tangent calculation:
- [tex]\( 31.230, 5.411, 28.828, 4.995 \)[/tex]
Thus, all the given options are mathematically consistent. The correct usage depends on what configuration of the right triangle's sides you view.
