Question

triangle abc has the given measures. solve the triangle(s), if any exist.
a = 162°, a = 6.1, b = 4
how many triangle(s) can possibly be formed?
1
complete
the measure of angle b is about °.
the measure of angle c is about °.
done

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}\). Substituting the given values \(A = 162^{\circ}\), \(a = 6.1\), and \(b = 4\) into the formula, we get \(\frac{6.1}{\sin162^{\circ}}=\frac{4}{\sin B}\).
First, calculate \(\sin162^{\circ}=\sin(180 - 18)^{\circ}=\sin18^{\circ}\approx0.3090\). Then, from \(\frac{6.1}{0.3090}=\frac{4}{\sin B}\), we can solve for \(\sin B\). Cross - multiplying gives \(6.1\sin B=4\times0.3090\), so \(\sin B=\frac{4\times0.3090}{6.1}\approx0.2026\).
Since \(A = 162^{\circ}\) is obtuse and \(a>b\), there is one triangle. And \(B=\sin^{- 1}(0.2026)\approx11.7^{\circ}\) (using a calculator in degree mode).

Step2: Find angle \(C\)

We know that the sum of the angles in a triangle is \(A + B+C=180^{\circ}\). So \(C = 180^{\circ}-A - B\). Substituting \(A = 162^{\circ}\) and \(B\approx11.7^{\circ}\), we get \(C=180-(162 + 11.7)=6.3^{\circ}\)

Answer:

The measure of angle \(B\) is about \(11.7^{\circ}\). The measure of angle \(C\) is about \(6.3^{\circ}\)