Question

how many triangles exist that fit the following criteria?

c = 65^{circ},c = 8,b = 5

Explanation:

Step1: Use the Law of Sines

The Law of Sines states that \(\frac{\sin B}{b}=\frac{\sin C}{c}\).
Substitute \(C = 65^{\circ}\), \(c = 8\), and \(b = 5\) into the formula: \(\sin B=\frac{b\sin C}{c}\).
So, \(\sin B=\frac{5\sin65^{\circ}}{8}\).
Calculate \(\sin65^{\circ}\approx0.9063\), then \(\sin B=\frac{5\times0.9063}{8}=\frac{4.5315}{8}\approx0.5664\).

Step2: Find angle \(B\)

Since \(\sin B\approx0.5664\), then \(B=\sin^{- 1}(0.5664)\approx34.5^{\circ}\) or \(B = 180^{\circ}-34.5^{\circ}=145.5^{\circ}\).
But \(A + B + C=180^{\circ}\). If \(B = 145.5^{\circ}\), then \(A=180^{\circ}-(145.5^{\circ}+65^{\circ})=180^{\circ}-210.5^{\circ}\), which is not possible.
So there is only one valid value for \(B\).

Answer:

\(1\)