Question

two sides and an angle are given. determine whether the given information results in one triangle, two triangles, or no triangle at all. solve any resulting triangle(s).
b = 5, c = 7, b = 10°
select the correct choice and, if necessary, fill in the answer boxes to complete your choice.
a. a single triangle is produced, where c ≈ □°, a ≈ □°, and a ≈ □
(type integers or decimals rounded to two decimal places as needed.)
b. two triangles are produced, where the triangle with the smaller angle c has c₁ ≈ □°, a₁ ≈ □°, and a₁ ≈ □, and the triangle with the larger angle c has c₂ ≈ □°, a₂ ≈ □°, and a₂ ≈ □
(type integers or decimals rounded to two decimal places as needed.)
c. no triangles are produced.

Explanation:

Step1: Use the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). We know \(b = 5\), \(c=7\), \(B = 10^{\circ}\). First, find \(\sin C\): \(\sin C=\frac{c\sin B}{b}\). Substitute the values: \(\sin C=\frac{7\sin10^{\circ}}{5}\). Calculate \(\sin10^{\circ}\approx0.1736\), then \(\sin C=\frac{7\times0.1736}{5}=\frac{1.2152}{5}=0.24304\). So \(C=\sin^{- 1}(0.24304)\approx14.1^{\circ}\) or \(C = 180^{\circ}-14.1^{\circ}=165.9^{\circ}\). But \(B = 10^{\circ}\), if \(C = 165.9^{\circ}\), then \(A=180^{\circ}-(10^{\circ}+165.9^{\circ})=4.1^{\circ}\), and using the Law of Sines \(\frac{a}{\sin A}=\frac{b}{\sin B}\), \(a=\frac{b\sin A}{\sin B}\). \(\sin4.1^{\circ}\approx0.0713\), \(a=\frac{5\times0.0713}{0.1736}\approx2.05\). If \(C = 14.1^{\circ}\), then \(A=180^{\circ}-(10^{\circ}+14.1^{\circ})=155.9^{\circ}\), \(\sin155.9^{\circ}\approx0.4089\), \(a=\frac{5\times0.4089}{0.1736}\approx11.81\)

Answer:

B. Two triangles are produced, where the triangle with the smaller angle \(C_1\approx14.1^{\circ}\), \(A_1\approx155.9^{\circ}\), and \(a_1\approx11.81\), and the triangle with the larger angle \(C_2\approx165.9^{\circ}\), \(A_2\approx4.1^{\circ}\), and \(a_2\approx2.05\)