Certainly! In order to determine the correct equation using the given value of [tex]\( b \)[/tex] to solve for [tex]\( a \)[/tex], we need to analyze each option and recall the tangent function definition in trigonometry.
The tangent function for an angle in a right-angled triangle is defined as:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
Here, [tex]\( \theta = 22.6^\circ \)[/tex].
Given the correct equation is of the form [tex]\(\tan(22.6^\circ) = \frac{a}{b}\)[/tex] with [tex]\(b = 13\)[/tex] (where [tex]\(b\)[/tex] is the adjacent side length and [tex]\(a\)[/tex] is the opposite side length), let's look at each option:
1. [tex]\(\tan \left(22.6^{\circ}\right) = \frac{a}{13}\)[/tex]:
This suggests that the tangent of [tex]\(22.6^\circ\)[/tex] is equal to the ratio of the length of the opposite side [tex]\(a\)[/tex] to the length of the adjacent side [tex]\(13\)[/tex].
2. [tex]\(\tan \left(22.6^{\circ}\right) = \frac{13}{a}\)[/tex]:
This would imply that the tangent of [tex]\(22.6^\circ\)[/tex] is equal to the ratio of the length of the adjacent side [tex]\(13\)[/tex] to the opposite side [tex]\(a\)[/tex], which does not align with the definition of the tangent function.
3. [tex]\(\tan \left(22.6^{\circ}\right) = \frac{a}{12}\)[/tex]:
This formula indicates that [tex]\(b\)[/tex] is [tex]\(12\)[/tex] instead of [tex]\(13\)[/tex], which contradicts the given information.
4. [tex]\(\tan \left(22.6^{\circ}\right) = \frac{12}{a}\)[/tex]:
This formula suggests [tex]\(b\)[/tex] is [tex]\(12\)[/tex] and swaps the relationship of opposite and adjacent, which is incorrect again both due to the given value and definition.
Thus, option 1 is the correct equation:
[tex]\[ \tan \left(22.6^{\circ}\right) = \frac{a}{13} \][/tex]
Now, solving for [tex]\(a\)[/tex]:
Given that:
[tex]\[ \tan \left(22.6^{\circ}\right) \approx 0.416 \][/tex]
The equation becomes:
[tex]\[ 0.416 = \frac{a}{13} \][/tex]
To find [tex]\(a\)[/tex], multiply both sides by [tex]\(13\)[/tex]:
[tex]\[ a = 13 \times 0.416 \][/tex]
[tex]\[ a = 5.408 \][/tex]
Thus, the correct form of the equation using the value of [tex]\(b\)[/tex] to solve for [tex]\(a\)[/tex] is:
[tex]\[ \boxed{\tan \left(22.6^{\circ}\right) = \frac{a}{13}} \][/tex]
And solving it gives:
[tex]\[ a \approx 5.408 \][/tex]
Therefore, option 1 is correct for the equation [tex]\(\tan \left(22.6^{\circ}\right) = \frac{a}{13}\)[/tex].
