To solve for [tex]\( b \)[/tex] and find the correct equation for [tex]\( a \)[/tex], follow these detailed steps:
1. Solve for [tex]\( b \)[/tex]:
Given:
[tex]\[
\cos(22.6^\circ) = \frac{b}{13}
\][/tex]
We need to isolate [tex]\( b \)[/tex]. Multiply both sides by 13:
[tex]\[
b = 13 \cdot \cos(22.6^\circ)
\][/tex]
We know from the given problem that this calculation yields:
[tex]\[
b \approx 12
\][/tex]
when rounded to the nearest whole number.
2. Identify the correct equation for [tex]\( a \)[/tex]:
To determine which equation correctly uses the value of [tex]\( b \)[/tex] to solve for [tex]\( a \)[/tex], we need to use the tangent function.
Since:
[tex]\[
\tan(22.6^\circ) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
In terms of the triangle [tex]\( \triangle ABC \)[/tex], where angle [tex]\( A \)[/tex] is [tex]\( 22.6^\circ \)[/tex], the sides are:
- Hypotenuse ([tex]\( c \)[/tex]) = 13
- Adjacent side to angle [tex]\( 22.6^\circ \)[/tex] ([tex]\( b \)[/tex]) = 12
- Opposite side to angle [tex]\( 22.6^\circ \)[/tex] ([tex]\( a \)[/tex]) = [tex]\( ? \)[/tex]
The tangent function states:
[tex]\[
\tan(22.6^\circ) = \frac{a}{b}
\][/tex]
Substitute [tex]\( b = 12 \)[/tex]:
[tex]\[
\tan(22.6^\circ) = \frac{a}{12}
\][/tex]
Therefore, the correct equation that uses the value of [tex]\( b \)[/tex] to solve for [tex]\( a \)[/tex] is:
[tex]\[
\tan \left(22.6^\circ\right) = \frac{a}{12}
\][/tex]
So the correct solution steps and the equation giving the value of [tex]\( a \)[/tex] are confirmed. The correct option is:
[tex]\[
\tan \left(22.6^\circ\right) = \frac{a}{12}
\][/tex]
