Question

assume α is opposite side a, β is opposite side b, and γ is opposite side c. determine whether there is no triangle, one triangle, or two triangles.
a = 4 b = 11 α = 48°
one triangle no triangles two triangles
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Explanation:

Step1: Use the Law of Sines

The Law of Sines is \(\frac{a}{\sin\alpha}=\frac{b}{\sin\beta}\). We know \(a = 4\), \(b=11\), \(\alpha = 48^{\circ}\). So \(\sin\beta=\frac{b\sin\alpha}{a}\).
Substitute the values: \(\sin\beta=\frac{11\sin48^{\circ}}{4}\).
Since \(\sin48^{\circ}\approx0.7431\), then \(\sin\beta=\frac{11\times0.7431}{4}=\frac{8.1741}{4}=2.0435\).

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

The range of the sine function is \([- 1,1]\). Here, \(\sin\beta = 2.0435>1\).

Answer:

No Triangles