Question

5 multiple choice 1 point use the law of sines to find the missing side of the triangle. find the measure of b, given ( mangle a = 51^{circ}, mangle b = 41^{circ} ), and ( a = 67 ). 79.4 34.2 56.6 69.8 6 multiple choice 1 point use the law of sines to find the missing side of the triangle. find the measure of b, given ( mangle a = 47^{circ}, mangle b = 24^{circ} ), and ( a = 54 ). 30.0 32.2 97.1 16.1

Explanation:

Step1: Apply the Law of Sines formula

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}\).

Step2: Substitute the given values

Given \(A = 51^{\circ}\), \(B = 41^{\circ}\), and \(a = 67\). Substitute into the formula: \(\frac{67}{\sin 51^{\circ}}=\frac{b}{\sin 41^{\circ}}\).

Step3: Solve for \(b\)

First, find \(\sin 51^{\circ}\approx0.777\) and \(\sin 41^{\circ}\approx0.656\). Then \(b=\frac{67\times\sin 41^{\circ}}{\sin 51^{\circ}}=\frac{67\times0.656}{0.777}\approx56.6\).

Step1: Apply the Law of Sines formula

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}\).

Step2: Substitute the given values

Given \(A = 47^{\circ}\), \(B = 24^{\circ}\), and \(a = 54\). Substitute into the formula: \(\frac{54}{\sin 47^{\circ}}=\frac{b}{\sin 24^{\circ}}\).

Step3: Solve for \(b\)

First, find \(\sin 47^{\circ}\approx0.731\) and \(\sin 24^{\circ}\approx0.407\). Then \(b=\frac{54\times\sin 24^{\circ}}{\sin 47^{\circ}}=\frac{54\times0.407}{0.731}\approx30.0\).

Answer:

\(56.6\)