Question

which equation can be used to solve for b?

triangle with right angle at c, angle at b is 30°, side bc is 8 ft, side ac is b, side ab is c

options:

• $b = (8)\tan(30^\circ)$
• $b = \frac{8}{\tan(30^\circ)}$
• $b = (8)\sin(30^\circ)$
• $b = \frac{8}{\sin(30^\circ)}$

Explanation:

Step1: Recall trigonometric ratio

In a right - triangle, for an acute angle \(\theta\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 30^{\circ}\), the adjacent side to \(30^{\circ}\) is \(b\), and the opposite side is \(b\), and the adjacent side to \(30^{\circ}\) is \(8\) (assuming the side of length \(8\) is adjacent to the \(30^{\circ}\) angle in the right - triangle \(ABC\) with \(\angle C = 90^{\circ}\)).

Step2: Apply the tangent formula

We know that \(\tan(30^{\circ})=\frac{b}{8}\) (by the definition of the tangent function \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\) in right - triangle \(ABC\) where \(\theta = 30^{\circ}\), opposite side to \(30^{\circ}\) is \(b\) and adjacent side is \(8\)).

Step3: Solve for \(b\)

Cross - multiply the equation \(\tan(30^{\circ})=\frac{b}{8}\) to get \(b = 8\tan(30^{\circ})\)

Answer:

\(b=(8)\tan(30^{\circ})\) (the first option)