Question

use the law of sines to find the value of a. what is the best approximation of the value of a? law of sines: $\frac{sin a}{a}=\frac{sin b}{b}=\frac{sin c}{c}$

Explanation:

1. First, find angle \(C\):

• In a triangle, the sum of the interior - angles is \(180^{\circ}\). Given \(A = 40^{\circ}\) and \(B = 95^{\circ}\), then \(C=180-(40 + 95)=45^{\circ}\).

1. Then, apply the Law of Sines:

• The Law of Sines states that \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\). We know \(b = 4.7\mathrm{cm}\), \(A = 40^{\circ}\), \(B = 95^{\circ}\), and \(C = 45^{\circ}\).
• Using \(\frac{\sin A}{a}=\frac{\sin B}{b}\), we can solve for \(a\). Cross - multiplying gives \(a=\frac{b\sin A}{\sin B}\).
• Substitute \(b = 4.7\mathrm{cm}\), \(\sin A=\sin40^{\circ}\approx0.6428\), and \(\sin B=\sin95^{\circ}\approx0.9962\) into the formula.
• \(a=\frac{4.7\times0.6428}{0.9962}=\frac{3.0212}{0.9962}\approx3.0\mathrm{cm}\).

Answer:

3.0 cm