When using the Law of Cosines, how do I know if two triangles exist?
To determine if two triangles exist using the Law of Cosines, you need to ensure that the given side lengths are suitable to form a triangle. The Law of Cosines states that for a triangle with side lengths a, b, and c, and opposite angles A, B, and C respectively, the equation is given by:
c^2 = a^2 + b^2 - 2ab * cos(C)
To determine if two triangles can be formed, you can use either the Angle-Side-Angle (ASA) or Side-Side-Side (SSS) criteria.
1. Angle-Side-Angle (ASA):
If you are given two angles and the included side, ensure that the following conditions are met:
- The sum of the two given angles is less than 180 degrees.
- The length of the included side is greater than the difference between the lengths of the other two sides and less than their sum.
2. Side-Side-Side (SSS):
If you are given the lengths of all three sides, ensure that the triangle inequality theorem is satisfied:
- The sum of the lengths of any two sides is always greater than the length of the third side.
By checking these criteria, you can determine if the given side lengths can form a valid triangle. If the conditions are satisfied, you can use the Law of Cosines to solve for the unknown angles or side lengths of the triangle.
You question makes no sense to me.
If it is a triangle, the law of cosines work.
There are cases where no triangle can exist with the given lengths or angles. For example if the sum of the two shorter sides is less than the long side or if the three angles do not add up to 180 degrees.