Question

what is the measure of angle b?

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). In \(\triangle ABC\), let \(a = 3.2\), \(b = 11\), and \(C=16.7^{\circ}\). First, find \(\sin A\) using \(\frac{a}{\sin A}=\frac{b}{\sin C}\). So, \(\sin A=\frac{a\sin C}{b}\). Substitute \(a = 3.2\), \(b = 11\), and \(C = 16.7^{\circ}\) (where \(\sin16.7^{\circ}\approx0.287\)). Then \(\sin A=\frac{3.2\times0.287}{11}\approx\frac{0.9184}{11}\approx0.0835\). So \(A\approx\sin^{- 1}(0.0835)\approx4.8^{\circ}\).

Step2: Use the angle - sum property of a triangle

Since the sum of angles in a triangle is \(180^{\circ}\), \(A + B+C=180^{\circ}\). Then \(B=180^{\circ}-(A + C)\). Substitute \(A\approx4.8^{\circ}\) and \(C = 16.7^{\circ}\). So \(B=180^{\circ}-(4.8^{\circ}+16.7^{\circ})=180^{\circ}-21.5^{\circ}=158.5^{\circ}\) (Wait, wrong! Re - check the Law of Sines application. The correct formula is \(\frac{\sin A}{a}=\frac{\sin C}{c}\) (mis - labeled sides above. Let \(AB = c = 3.2\), \(AC=b = 11\), \(\angle B\) is opposite to \(AC\), \(\angle C = 16.7^{\circ}\) opposite to \(AB\)). By the Law of Sines \(\frac{\sin B}{b}=\frac{\sin C}{c}\). So \(\sin B=\frac{b\sin C}{c}\). Substitute \(b = 11\), \(c = 3.2\), \(\sin C=\sin16.7^{\circ}\approx0.287\). Then \(\sin B=\frac{11\times0.287}{3.2}=\frac{3.157}{3.2}\approx0.9866\). So \(B=\sin^{-1}(0.9866)\approx81.04^{\circ}\)

Answer:

\(81.04^{\circ}\)