Trigonometry is typically taught in geometry or algebra II classes.
The Law of Sines states that the sine of an angle divided by the length of the side opposite that angle yields the same quotient for all three angles of a triangle. Written in equation form, the Law of Sines is: (sin A)/a = (sin B)/b = (sin C)/c, where A, B and C are the three angles of the triangle and a, b and c are the three sides. The Law of Sines can be used in the Ambiguous Case (when two side lengths are known in addition to one angle that is not between the two sides).
Instructions
Determine the Number of Solutions
1. Find the altitude of the triangle, which is equal to the side adjacent to the given angle times the sine of the given angle.
2. Compare the altitude of the triangle to the side opposite the given angle. If the altitude is greater than the side opposite the given angle, there is no solution to the triangle. If the altitude is equal to the side opposite the given angle, there is one solution to the triangle.
3. Compare the side adjacent to the given angle to the side opposite the given angle. If the side opposite the given angle is greater than the side adjacent to the given angle, there is one solution to the triangle. If the altitude is less than the side opposite the given angle and the side opposite the given angle is less than the side adjacent to the given angle, there are two solutions to the triangle.
Solve the Triangle(s)
4. Set up the Law of Sines for two angles: (sin A)/a = (sin B)/b.
5. Plug in any known values that you have (two side lengths and one angle). At this point, the only variable left should be either A or B.
6. Use a scientific calculator to evaluate the expression and solve for the unknown angle.
7. Subtract the sum of the two known angles from 180 to obtain the third angle. For two-solution problems, the second angle solution is equal to 180 minus the angle solution that you just calculated.
8. Use the Law of Sines to solve for the third side. Do this once for one-solution problems and twice for two-solution problems.
Tags: given angle, side opposite, opposite given, opposite given angle, side opposite given, adjacent given