How do you solve a triangle if you are given A = 120 degrees, b = 3, c = 10?

To solve a triangle when given one angle and two sides, you can use the Law of Sines and Law of Cosines. Here's how to proceed:

1. First, use the Law of Sines to find the ratio of the lengths of the sides to the sines of their opposite angles. This law states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.

   [ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} ]

2. Given angle ( A = 120^\circ ), and sides ( b = 3 ) and ( c = 10 ), you can use the Law of Sines to find the ratio of the lengths of the sides:

   [ \frac{a}{\sin(120^\circ)} = \frac{3}{\sin(B)} = \frac{10}{\sin(C)} ]

3. Solve for the unknown angles using the given side lengths and the known angle.

4. Once you have found the measures of angles ( B ) and ( C ), you can use the fact that the sum of angles in a triangle is ( 180^\circ ) to find angle ( A ).

5. Use the Law of Cosines to find the remaining side length. The Law of Cosines states that for any triangle with side lengths ( a ), ( b ), and ( c ), and opposite angles ( A ), ( B ), and ( C ), respectively:

   [ c^2 = a^2 + b^2 - 2ab \cos(C) ]

6. Plug in the known values to find the length of side ( a ).

7. Once you have found all the side lengths and angles, you have solved the triangle.