Question

use the law of sines to find the indicated side length in this triangle. ? 13 96° 25° round your answer to the nearest hundredth.

Explanation:

Step1: Find the third angle

The sum of angles in a triangle is \(180^\circ\). Let the third angle be \(C\). So, \(C = 180^\circ - 96^\circ - 25^\circ = 59^\circ\).

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Let the side opposite \(96^\circ\) be \(a\) (the unknown), the side opposite \(25^\circ\) be \(13\) (let's call it \(b\)), and the angle opposite \(13\) is \(25^\circ\), angle opposite \(a\) is \(96^\circ\). So, \(\frac{a}{\sin 96^\circ}=\frac{13}{\sin 25^\circ}\).

Step3: Solve for \(a\)

Rearrange the formula: \(a=\frac{13\times\sin 96^\circ}{\sin 25^\circ}\). Calculate \(\sin 96^\circ\approx0.9945\) and \(\sin 25^\circ\approx0.4226\). Then \(a = \frac{13\times0.9945}{0.4226}\approx\frac{12.9285}{0.4226}\approx30.59\).

Answer:

\(30.59\)