Question

use the law of sines to find the missing angle of the triangle. find ( mangle b ) to the nearest tenth. examples: 1) find ( mangle c ) ( \frac{sin 89^{circ}}{33}=sin )

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\). In the given triangle, \(a = 75\), \(A=29^{\circ}\), and \(b = 154\). We want to find \(B\). So, \(\frac{\sin B}{75}=\frac{\sin29^{\circ}}{154}\).

Step2: Solve for \(\sin B\)

Cross - multiply: \(\sin B=\frac{75\times\sin29^{\circ}}{154}\).
We know that \(\sin29^{\circ}\approx0.4848\). Then \(\sin B=\frac{75\times0.4848}{154}=\frac{36.36}{154}\approx0.2361\).

Step3: Find the angle \(B\)

Since \(B = \sin^{- 1}(0.2361)\), using a calculator, \(B\approx13.7^{\circ}\).

Answer:

\(13.7^{\circ}\)