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Ambiguous case of the Law of sines
Explained with 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.
Triangle 2
Use the formula for the law of sines to determine the unknown side length.
As you can see this second triangle
The picture below shows both triangles and illustrates this example of the ambiguous case of the law of sines. 
Genereal 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
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