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).
a = 6, b = 2, a = 80°
select the correct choice and, if necessary, fill in the answer boxes to complete your choice.
a. a single triangle is produced, where b ≈ □°, c ≈ □°, and c ≈ 6.01
(type integers or decimals rounded to two decimal places as needed.)
b. two triangles are produced, where the triangle with the smaller angle b has b₁ ≈ □°, c₁ ≈ □°, and c₁ ≈ □; and the triangle with the larger angle b has b₂ ≈ □°, c₂ ≈ □°, and c₂ ≈ □
(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 \(a = 6\), \(b = 2\), and \(A=80^{\circ}\). So, \(\frac{6}{\sin80^{\circ}}=\frac{2}{\sin B}\). Then \(\sin B=\frac{2\sin80^{\circ}}{6}\).
Calculate \(\sin80^{\circ}\approx0.9848\), so \(\sin B=\frac{2\times0.9848}{6}\approx0.3283\). Then \(B=\sin^{- 1}(0.3283)\approx19.18^{\circ}\).

Step2: Find angle \(C\)

Since the sum of angles in a triangle is \(180^{\circ}\), \(C = 180^{\circ}-A - B\). Substitute \(A = 80^{\circ}\) and \(B\approx19.18^{\circ}\), so \(C=180^{\circ}-80^{\circ}-19.18^{\circ}=80.82^{\circ}\).

Step3: Find side \(c\)

Using the Law of Sines again, \(\frac{c}{\sin C}=\frac{a}{\sin A}\). So \(c=\frac{a\sin C}{\sin A}\). Substitute \(a = 6\), \(\sin A=\sin80^{\circ}\approx0.9848\), \(\sin C=\sin80.82^{\circ}\approx0.9869\). Then \(c=\frac{6\times0.9869}{0.9848}\approx6.01\).

Answer:

A. A single triangle is produced, where \(B\approx19.18^{\circ}\), \(C\approx80.82^{\circ}\), and \(c\approx6.01\)