Question

which equation can be used to solve for b?
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: Identify the triangle type and trigonometric ratios

We have a right - triangle \( \triangle ABC \) with \( \angle C = 90^{\circ} \), \( \angle B=30^{\circ} \), and the length of side \( BC = 8\) ft (adjacent to \( \angle B \)), and we need to find the length of side \( b \) (opposite to \( \angle B \)).

In a right - triangle, the tangent of an angle \( \theta \) is defined as the ratio of the length of the opposite side to the length of the adjacent side. The formula for tangent is \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \).

Step2: Apply the tangent ratio to the given triangle

For \( \angle B = 30^{\circ} \), the opposite side to \( \angle B \) is \( b \) and the adjacent side to \( \angle B \) is \( BC = 8 \) ft.

Using the tangent formula \( \tan(30^{\circ})=\frac{b}{8} \).

To solve for \( b \), we can multiply both sides of the equation by 8: \( b = 8\times\tan(30^{\circ}) \)

Answer:

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