Question

use the triangle shown on the right to answer the question. which equation can you use to solve for a?
\\( \frac { a } { \sin ( 35 ^ { \circ } ) } = \frac { 30 } { \sin ( 35 ^ { \circ } ) } \\)
\\( \frac { a } { \sin ( 35 ^ { \circ } ) } = \frac { 30 } { \sin ( 60 ^ { \circ } ) } \\)
\\( \frac { a } { \sin ( 85 ^ { \circ } ) } = \frac { 30 } { \sin ( 35 ^ { \circ } ) } \\)

Explanation:

Step1: Recall the Law of Sines

The Law of Sines states that in a triangle \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\).

Step2: Identify the sides and angles

In the given triangle, side \(a\) is opposite an angle of \(85^{\circ}\) (since the sum of angles in a triangle is \(180^{\circ}\), \(B = 180^{\circ}-35^{\circ}-85^{\circ}=60^{\circ}\) is not relevant here as we are relating \(a\) and the side of length \(30\) which is opposite the \(35^{\circ}\) angle. So using the Law of Sines \(\frac{a}{\sin(85^{\circ})}=\frac{30}{\sin(35^{\circ})}\).

Answer:

\(\frac{a}{\sin(85^{\circ})}=\frac{30}{\sin(35^{\circ})}\) (the third option in the given choices)