Question

solve for b.
law of sines: $\frac{sin a}{a} = \frac{sin b}{b} = \frac{sin c}{c}$
round your answer to the nearest hundredth.
(there is a triangle with vertices b, a, c. angle at b is 27°, angle at c is 40°, side c (opposite angle c) is 15, side b (opposite angle b) is to be found.)

Explanation:

Step1: Find angle A

In a triangle, the sum of angles is \(180^\circ\). So, \( \angle A = 180^\circ - 27^\circ - 40^\circ = 113^\circ \).

Step2: Apply Law of Sines

Law of Sines: \( \frac{\sin B}{b} = \frac{\sin C}{c} \). We know \( \angle B = 27^\circ \), \( \angle C = 40^\circ \), \( c = 15 \). Plug in values: \( \frac{\sin 27^\circ}{b} = \frac{\sin 40^\circ}{15} \).

Step3: Solve for b

Rearrange formula: \( b = \frac{15 \times \sin 27^\circ}{\sin 40^\circ} \). Calculate \( \sin 27^\circ \approx 0.4540 \), \( \sin 40^\circ \approx 0.6428 \). Then \( b = \frac{15 \times 0.4540}{0.6428} \approx \frac{6.81}{0.6428} \approx 10.59 \).

Answer:

\( b \approx 10.59 \)