Question

solve for a.
(there is a triangle with side a, side 13, angle 68° and angle 83°; a = ?; round your final answer to the nearest tenth.
law of sines: $\frac{a}{sin a} = \frac{b}{sin b} = \frac{c}{sin c}$)

Explanation:

Step 1: Find the third angle

The sum of angles in a triangle is \(180^\circ\). Let the third angle be \(C\). So, \(C = 180^\circ - 68^\circ - 83^\circ = 29^\circ\).

Step 2: Apply the Law of Sines

We know that \(\frac{a}{\sin A}=\frac{c}{\sin C}\), where \(c = 13\), \(A = 83^\circ\), and \(C = 29^\circ\). So, \(a=\frac{13\times\sin83^\circ}{\sin29^\circ}\).
First, calculate \(\sin83^\circ\approx0.9925\) and \(\sin29^\circ\approx0.4848\).
Then, \(a=\frac{13\times0.9925}{0.4848}=\frac{12.9025}{0.4848}\approx26.6\).

Answer:

\(26.6\)