Question

find the angle b. do not use a calculator.
b = \square ^ { \circ }

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Let \(a = 6\sqrt{3}\), \(A = 60^{\circ}\), \(b = 6\). Then \(\frac{6\sqrt{3}}{\sin60^{\circ}}=\frac{6}{\sin B}\).
Since \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\), we substitute it into the equation: \(\frac{6\sqrt{3}}{\frac{\sqrt{3}}{2}}=\frac{6}{\sin B}\).
Simplify the left - hand side: \(\frac{6\sqrt{3}}{\frac{\sqrt{3}}{2}}=6\sqrt{3}\times\frac{2}{\sqrt{3}} = 12\). So, \(12=\frac{6}{\sin B}\).

Step2: Solve for \(\sin B\)

From \(12=\frac{6}{\sin B}\), we can cross - multiply to get \(12\sin B=6\). Then \(\sin B=\frac{6}{12}=\frac{1}{2}\).

Step3: Determine the angle \(B\)

Since \(b = 6\lt a = 6\sqrt{3}\), angle \(B\) is acute. And we know that if \(\sin B=\frac{1}{2}\) and \(B\) is acute, then \(B = 30^{\circ}\).

Answer:

\(30\)