Solve the following triangle. \[ B=20^{\circ}, C=60^{\circ}, b=4 \]
Solution
Step 1 ：Given that in a triangle, the sum of the angles is \(180^{\circ}\), we can find angle A by subtracting the sum of angles B and C from \(180^{\circ}\). So, \(A = 180^{\circ} - B - C = 180^{\circ} - 20^{\circ} - 60^{\circ} = 100^{\circ}\).
Step 2 ：Next, we use the Law of Sines to find the lengths of sides a and c. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. So, we can set up the following equations to find a and c: \(a = b \cdot \frac{\sin A}{\sin B}\) and \(c = b \cdot \frac{\sin C}{\sin B}\).
Step 3 ：Substituting the given values into the equations, we get \(a = 4 \cdot \frac{\sin 100^{\circ}}{\sin 20^{\circ}} \approx 11.52\) and \(c = 4 \cdot \frac{\sin 60^{\circ}}{\sin 20^{\circ}} \approx 10.13\).
Step 4 ：\(\boxed{A = 100^{\circ}, a = 11.52, c = 10.13}\) is the solution to the triangle.
