The ambiguous case of the law of sines occurs when two different angle measurements satisfy the formula for the law of sines .

**Video** Tutorial on the Ambiguous Case

### Examples

Example of 2 triangles satisfying the law of sines formula

Consider the following problem. A triangle has side lengths of 20 and 11. The angle opposite the latter side is 29° . What is the measure of the angle opposite of the side that has a length of 20? See the picture immediately below.

Triangle 1

Use the law of sines formula to find the measure of x in the triangle below.

There is actually a 2nd triangle that satisfies the law of sines for this problem.

.The picture below shows both triangles and illustrates this example of the ambiguous case of the law of sines.

##### Interactive Demonstration of the Ambiguous Case

### General Steps for solving the ambiguous case

- You will be given two sides and 1 angle. Use these and the formula for the law of sines to get the second angle
- Check if the first angle that you got is valid
- ie Do the two angles that you currently have a sum that is less than 180

- Check if there is a second angle that is valid. To do this, find the angle in Quadrant II that has the same sine as the angle that you found in Step 2 (subtract that angle from 180)
- If the angle in Quadrant II plus the original angle you were given have a sum that is less than 180, you have a second triangle that works.
- Try the interactive demonstration below to better understand how to use these steps to solve the ambiguous case of the law of sines