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?

Explanation:

Step1: Use the Law of Sines

The Law of Sines is \(\frac{a}{\sin A}=\frac{b}{\sin B}\). Substitute \(A = 162^{\circ}\), \(a = 6.1\), and \(b = 4\) into the formula: \(\sin B=\frac{b\sin A}{a}\).

Step2: Calculate \(\sin B\)

\(\sin B=\frac{4\times\sin(162^{\circ})}{6.1}\). Since \(\sin(162^{\circ})=\sin(180 - 18)^{\circ}=\sin18^{\circ}\approx0.3090\), then \(\sin B=\frac{4\times0.3090}{6.1}\approx\frac{1.236}{6.1}\approx0.2026\).

Step3: Analyze the value of \(B\)

Since \(A = 162^{\circ}\) (an obtuse angle) and \(a>b\) (in a triangle, if \(A\) is obtuse and \(a > b\), there is exactly one triangle). Also, from \(\sin B\approx0.2026\), \(B=\sin^{- 1}(0.2026)\approx11.67^{\circ}\) (the other value of \(B\) from \(\sin B = 0.2026\) is \(180^{\circ}-11.67^{\circ}=168.33^{\circ}\), but \(A + 168.33^{\circ}=162^{\circ}+168.33^{\circ}=330.33^{\circ}>180^{\circ}\), so we discard it).

Answer:

One triangle can be formed.