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 = 6 b = 7 α = 43°
no triangles two triangles one triangle
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Explanation:

Step1: Use the Law of Sines

By the Law of Sines, \(\frac{\sin\beta}{b}=\frac{\sin\alpha}{a}\). Substitute \(a = 6\), \(b = 7\), \(\alpha=43^{\circ}\). So \(\sin\beta=\frac{b\sin\alpha}{a}=\frac{7\sin43^{\circ}}{6}\).
Calculate \(\sin43^{\circ}\approx0.682\), then \(\sin\beta=\frac{7\times0.682}{6}=\frac{4.774}{6}\approx0.796\).

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

Since \(0\lt\sin\beta\lt1\), and \(b\gt a\), there are two possible values for \(\beta\) (one acute \(\beta_1=\sin^{- 1}(0.796)\approx52.8^{\circ}\) and one obtuse \(\beta_2 = 180^{\circ}-\beta_1\approx127.2^{\circ}\)) such that the sum of angles in a triangle \(\alpha+\beta+\gamma = 180^{\circ}\) is satisfied.

Answer:

Two Triangles