Question

in the triangle below, b = ______. if necessary, round your answer to two decimal places. (the triangle has angle at a: 46.7°, angle at c: 48°, side bc: 44, side b is opposite angle b, side c is opposite angle c)

Explanation:

Step1: Find angle B

The sum of angles in a triangle is \(180^\circ\). So, \( \angle B = 180^\circ - 46.7^\circ - 48^\circ \)
\( \angle B = 85.3^\circ \)

Step2: Apply the Law of Sines

The Law of Sines states that \( \frac{b}{\sin B} = \frac{BC}{\sin A} \). Here, \( BC = 44 \), \( \angle A = 46.7^\circ \), \( \angle B = 85.3^\circ \)
So, \( b = \frac{44 \times \sin 85.3^\circ}{\sin 46.7^\circ} \)

Step3: Calculate the sines

\( \sin 85.3^\circ \approx 0.9965 \), \( \sin 46.7^\circ \approx 0.7280 \)

Step4: Compute b

\( b = \frac{44 \times 0.9965}{0.7280} \approx \frac{43.846}{0.7280} \approx 60.23 \)

Answer:

\( 60.23 \)