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 12°. the measure of angle c is about 6°. complete what is the approximate side length of c? 2 4 12 18 done

Explanation:

Step1: Use the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). We know \(A = 162^{\circ}\), \(a = 6.1\), \(b = 4\), and \(B\approx12^{\circ}\), \(C\approx6^{\circ}\).

Step2: Apply the Law of Sines for side \(c\)

Since \(\frac{a}{\sin A}=\frac{c}{\sin C}\), we can solve for \(c\). Substitute the values: \(c=\frac{a\sin C}{\sin A}\).

$$ c=\frac{6.1\times\sin(6^{\circ})}{\sin(162^{\circ})} $$

We know that \(\sin(162^{\circ})=\sin(180 - 18)^{\circ}=\sin(18^{\circ})\) and \(\sin(6^{\circ})\approx0.1045\), \(\sin(18^{\circ})\approx0.3090\)

$$ c=\frac{6.1\times0.1045}{0.3090}=\frac{0.63745}{0.3090}\approx2 $$

Answer:

2