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Which equation can be used to solve for [tex]b[/tex]?

A. [tex]\tan \left(30^{\circ}\right)=\frac{5}{b}[/tex]

B. [tex]\tan (3 \pi x)=\frac{b}{5}[/tex]

Sagot :

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]