Question

using the law of sines to solve the all possible triangles if \\(\angle a = 117^\circ, a = 34, b = 18\\). if no answer exists, enter dne for all answers. \\(\angle b\\) is \\(\boxed{}\\) degrees \\(\angle c\\) is \\(\boxed{}\\) degrees \\(c = \boxed{}\\) assume \\(\angle a\\) is opposite side \\(a\\), \\(\angle b\\) is opposite side \\(b\\), and \\(\angle c\\) is opposite side \\(c\\). question help: \\(\boxed{}\\) video

Explanation:

Step1: Apply the Law of Sines to find \(\angle B\)

The Law of Sines states that \(\frac{\sin A}{a}=\frac{\sin B}{b}\).
Substituting the given values \(A = 117^{\circ}\), \(a = 34\), and \(b = 18\) into the formula:
\(\sin B=\frac{b\sin A}{a}=\frac{18\sin117^{\circ}}{34}\)
\(\sin117^{\circ}=\sin(180 - 63)^{\circ}=\sin63^{\circ}\approx0.891\)
\(\sin B=\frac{18\times0.891}{34}\approx\frac{16.038}{34}\approx0.4717\)
\(B=\sin^{- 1}(0.4717)\approx28.1^{\circ}\)

Step2: Find \(\angle C\)

Since the sum of angles in a triangle is \(180^{\circ}\), \(C = 180^{\circ}-A - B\)
\(C=180^{\circ}-117^{\circ}-28.1^{\circ}=34.9^{\circ}\)

Step3: Use the Law of Sines to find \(c\)

Again, by the Law of Sines \(\frac{\sin A}{a}=\frac{\sin C}{c}\)
\(c=\frac{a\sin C}{\sin A}\)
\(\sin C=\sin34.9^{\circ}\approx0.572\), \(\sin A=\sin117^{\circ}\approx0.891\)
\(c=\frac{34\times0.572}{0.891}\approx\frac{19.448}{0.891}\approx21.8\)

Answer:

\(\angle B\) is \(28.1\) degrees, \(\angle C\) is \(34.9\) degrees, \(c = 21.8\)