To solve for [tex]\( b \)[/tex] in the given equation, let's examine both equations step-by-step:
First equation:
[tex]\[
\tan \left(30^\circ\right) = \frac{5}{b}
\][/tex]
1. To isolate [tex]\( b \)[/tex], we can multiply both sides of the equation by [tex]\( b \)[/tex]:
[tex]\[
b \cdot \tan \left(30^\circ\right) = 5
\][/tex]
2. Next, divide both sides by [tex]\(\tan \left(30^\circ\right)\)[/tex] to solve for [tex]\( b \)[/tex]:
[tex]\[
b = \frac{5}{\tan \left(30^\circ\right)}
\][/tex]
This equation allows us to find [tex]\( b \)[/tex] directly when given [tex]\(\tan \left(30^\circ\right)\)[/tex].
Second equation:
[tex]\[
\tan(3 \pi x) = \frac{b}{5}
\][/tex]
1. To solve for [tex]\( b \)[/tex], multiply both sides by 5:
[tex]\[
5 \cdot \tan(3 \pi x) = b
\][/tex]
While this equation also solves for [tex]\( b \)[/tex], it involves an additional variable [tex]\( x \)[/tex], which is not being defined in the question. Therefore, the first equation is preferred because it directly relates [tex]\( b \)[/tex] and a known trigonometric value.
Choice of equation:
Based on the clarity and direct relationship between [tex]\( b \)[/tex] and the known value [tex]\(\tan \left(30^\circ\right)\)[/tex], the correct equation to use to solve for [tex]\( b \)[/tex] is:
[tex]\[
\tan \left(30^\circ\right) = \frac{5}{b}
\][/tex]
Thus, the correct answer is:
[tex]\[
\tan \left(30^\circ\right) = \frac{5}{b}
\][/tex]
