Question

directions: use the law of sines to find each missing side or angle. round to the nearest tenth.
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Explanation:

Let's solve problem 2 using the Law of Sines.

The Law of Sines states that for any triangle, \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\), where \(a, b, c\) are the lengths of the sides opposite angles \(A, B, C\) respectively.

Step 1: Find the third angle

First, we find the third angle of the triangle. The sum of angles in a triangle is \(180^{\circ}\). Given angles are \(65^{\circ}\) and \(53^{\circ}\), so the third angle \(\theta=180-(65 + 53)=180 - 118 = 62^{\circ}\)

Step 2: Apply the Law of Sines

We know that the side of length \(22\) is opposite the angle \(62^{\circ}\) and the side \(x\) is opposite the angle \(53^{\circ}\). Using the Law of Sines \(\frac{x}{\sin53^{\circ}}=\frac{22}{\sin62^{\circ}}\)

We can solve for \(x\) by cross - multiplying:
\(x=\frac{22\times\sin53^{\circ}}{\sin62^{\circ}}\)

We know that \(\sin53^{\circ}\approx0.7986\) and \(\sin62^{\circ}\approx0.8829\)

\(x=\frac{22\times0.7986}{0.8829}=\frac{17.5692}{0.8829}\approx19.9\)

Answer:

\(x\approx19.9\)