Question

solve for b.

(there is a triangle with angles 27°, 40°, and side 15 opposite to 40° angle, side b opposite to 27° angle)

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 \(C\). So, \(C = 180^\circ - 27^\circ - 40^\circ = 113^\circ\).

Step2: Apply the Law of Sines

We know that \(\frac{a}{\sin A}=\frac{b}{\sin B}\). Here, \(a = 15\), \(A = 113^\circ\), \(B = 27^\circ\), and we need to find \(b\).
So, \(\frac{15}{\sin 113^\circ}=\frac{b}{\sin 27^\circ}\).
First, calculate \(\sin 113^\circ\approx\sin(90^\circ + 23^\circ)=\cos 23^\circ\approx0.9205\) and \(\sin 27^\circ\approx0.4540\).
Then, \(b=\frac{15\times\sin 27^\circ}{\sin 113^\circ}\).
Substitute the values: \(b=\frac{15\times0.4540}{0.9205}\).
Calculate numerator: \(15\times0.4540 = 6.81\).
Then, \(b=\frac{6.81}{0.9205}\approx7.4\).

Answer:

\(7.4\)