Question

solve for b.

round your final answer to the nearest tenth.

law of sines: \\( \frac { a } { \sin a } = \frac { b } { \sin b } = \frac { c } { \sin c } \\)

Explanation:

Step1: Find the third angle

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle be \(A\). Then \(A = 180^{\circ}-27^{\circ}-40^{\circ}=113^{\circ}\).

Step2: Apply the Law of Sines

By the Law of Sines \(\frac{a}{\sin A}=\frac{b}{\sin B}\). Here \(a = 15\), \(A = 113^{\circ}\), \(B=27^{\circ}\). So \(\frac{15}{\sin113^{\circ}}=\frac{b}{\sin27^{\circ}}\).
Cross - multiply: \(b=\frac{15\times\sin27^{\circ}}{\sin113^{\circ}}\).
We know that \(\sin113^{\circ}=\sin(90^{\circ} + 23^{\circ})=\cos23^{\circ}\approx0.9205\), \(\sin27^{\circ}\approx0.4540\).
Then \(b=\frac{15\times0.4540}{0.9205}=\frac{6.81}{0.9205}\approx7.4\).

Answer:

\(7.4\)