Question

3 multiple choice 1 point the given measurements may or may not determine a triangle. if not, then state that no triangle is formed. if a triangle is formed, then use the law of sines to solve the triangle, if it is possible, or state that the law of sines cannot be used. b = 127°, c = 4, b = 9 no triangle is formed. c = 32.2°, a = 20.8°, a = 6 c = 20.8°, a = 32.2°, a = 6 the triangle cannot be solved with the law of sines.

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{\sin B}{b}=\frac{\sin C}{c}\).
Substitute \(B = 127^{\circ}\), \(b = 9\), and \(c = 4\) into the formula: \(\sin C=\frac{c\sin B}{b}\).
\(\sin C=\frac{4\sin127^{\circ}}{9}\).
Since \(\sin127^{\circ}=\sin(180 - 53)^{\circ}=\sin53^{\circ}\approx0.7986\), then \(\sin C=\frac{4\times0.7986}{9}\approx0.355\).
\(C=\sin^{- 1}(0.355)\approx20.8^{\circ}\).

Step2: Find angle \(A\)

Since the sum of angles in a triangle is \(A + B + C=180^{\circ}\), then \(A = 180^{\circ}-B - C\).
Substitute \(B = 127^{\circ}\) and \(C = 20.8^{\circ}\), \(A=180-(127 + 20.8)=32.2^{\circ}\).

Step3: Find side \(a\)

Again, using the Law of Sines \(\frac{\sin A}{a}=\frac{\sin B}{b}\).
\(a=\frac{b\sin A}{\sin B}\).
Substitute \(b = 9\), \(A = 32.2^{\circ}\), \(B = 127^{\circ}\).
\(\sin32.2^{\circ}\approx0.533\), \(\sin127^{\circ}\approx0.7986\).
\(a=\frac{9\times0.533}{0.7986}\approx6\).

Answer:

C. \(C = 20.8^{\circ},A = 32.2^{\circ},a = 6\)