Question

question 9 (mandatory) (1 point) determine \\( \angle b \\) to the nearest degree for the triangle with the given information \\( a = 3.7 \mathrm{~m}, b = 4.9 \mathrm{~m}, \angle a = 46^{circ} \\) a) \\( \angle b = 72^{circ} \\) b) \\( \angle b = 59^{circ} \\) c) \\( \angle b = 53^{circ} \\) d) \\( \angle b = 61^{circ} \\)

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}\).
Substitute \(a = 3.7\), \(b = 4.9\), and \(A=46^{\circ}\) into the formula: \(\frac{3.7}{\sin46^{\circ}}=\frac{4.9}{\sin B}\).

Step2: Solve for \(\sin B\)

Cross - multiply to get \(3.7\sin B=4.9\sin46^{\circ}\).
First, calculate \(\sin46^{\circ}\approx0.7193\). Then \(3.7\sin B = 4.9\times0.7193\).
\(3.7\sin B=3.5246\).
\(\sin B=\frac{3.5246}{3.7}\approx0.9526\).

Step3: Find the angle \(B\)

Since \(\sin B\approx0.9526\), \(B=\sin^{- 1}(0.9526)\).
Using a calculator, \(B\approx72^{\circ}\) (we consider the acute angle solution as in the context of triangle angle sum \(A + B+C=180^{\circ}\) and typical triangle problems without ambiguity indicated otherwise).

Answer:

a) \(\angle B = 72^{\circ}\)