Question

how many triangles exist that fit the following criteria?
c = 75°, c = 3, b = 5

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{\sin B}{b}=\frac{\sin C}{c}\). Substituting the given values \(C = 75^{\circ}\), \(c = 3\), and \(b = 5\), we get \(\sin B=\frac{b\sin C}{c}\).

$$ \sin B=\frac{5\sin75^{\circ}}{3} $$

Since \(\sin75^{\circ}=\sin(45^{\circ} + 30^{\circ})=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.966\)

$$ \sin B=\frac{5\times0.966}{3}\approx1.61 $$

Step2: Analyze the value of \(\sin B\)

The range of the sine function is \([- 1,1]\). Since \(\sin B\approx1.61>1\), there is no angle \(B\) for which this equation holds.

Answer:

\(0\)