Ambiguous case of the Law of sines

Explained with examples

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.

ambiguous case of law of sines
ambiguous case of law of sines

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

.
As you can see this second triangle below
ambiguous case of law of sines

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

picture of the ambiguous case of law of sines
Interactive Demonstration of the Ambiguous Case

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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

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