Question

problem #2: solve triangle abc if a = 65°, b = 65°, and c = 6. if no triangle exists, explain why.

Explanation:

Step1: Find angle C

The sum of angles in a triangle is \(180^\circ\). So, \(C = 180^\circ - A - B\).
Substituting \(A = 65^\circ\) and \(B = 65^\circ\), we get \(C = 180^\circ - 65^\circ - 65^\circ = 50^\circ\).

Step2: Use the Law of Sines to find sides a and b

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\).
First, find side a:
\(\frac{a}{\sin 65^\circ}=\frac{6}{\sin 50^\circ}\)
\(a=\frac{6\sin 65^\circ}{\sin 50^\circ}\)
Calculating \(\sin 65^\circ\approx0.9063\) and \(\sin 50^\circ\approx0.7660\), we have \(a=\frac{6\times0.9063}{0.7660}\approx\frac{5.4378}{0.7660}\approx7.1\) (rounded to one decimal place).

Then, find side b:
Since \(A = B = 65^\circ\), the triangle is isosceles with \(a = b\). So, \(b\approx7.1\) as well.

Answer:

Angle \(C = 50^\circ\), side \(a\approx7.1\), side \(b\approx7.1\), side \(c = 6\).