The Ambiguous Case of the Law of Sines

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  • 0:05 The Law of Sines
  • 0:51 The Ambiguous Case
  • 1:29 But Be Careful
  • 2:11 Example
  • 4:10 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

You might think that all math in very concrete, that an answer or a law is always right. But this is not always so. See, in certain situations, the law of sines actually give you two answers.

The Law of Sines

Ambiguity is a very interesting thing in math. You wouldn't think that a math law would give you two different answers to the same problem. But sometimes it does. In this lesson, we take a look at this odd ambiguity that results when certain conditions are met when we use the law of sines, which is very useful for solving problems involving triangles.

The law of sines tells us how the sides of the triangle are related to the angles of the triangle. Our triangle has sides a, b and c with angles A, B and C. Angle A is opposite side a, angle B is opposite side b, and angle C is opposite side c. Remember: side over the sine of the opposite angle. The law of sines works for any triangle.

The Ambiguous Case

But it doesn't work in all cases. When you are given two adjacent sides of a triangle followed by an angle, the law of sines will actually give you two answers. We call this situation the side-side-angle case. You can picture these two possibilities by swinging the side between the two unknown angles to see if you can form another triangle. For our triangle, if we swung side a to the left, we would see another triangle form when it touches the unknown side again. So, we have a possible obtuse triangle and a possible acute triangle.

But Be Careful

But we have to be careful when we apply the law of sines. See, the law of sines will only give us one of these angles if we are using a calculator. To find the other one, we need to subtract our calculator answer from 180. And even though we have two answers, there still may only be just the one answer. You have to visually check like we did above. Or you can look at our two answers and see which one works with our known angle. If both angles work, meaning they don't add up to more than 180 with our known triangle, then we have two possible answers. But if one of the angles adds up to 180 or more with our known angle, then we only have one possible answer.


Let's look at an example. Find the measure of angle x. We need to find the measure of angle x, we are given one angle that measures 33 and two sides that measure 4.7 and 6.3. To find the measure of angle x, we need to use the law of sines since our triangle is not a special triangle, like a right triangle. We can first label our sides and their corresponding opposite angles. We can label the 4.7 as side a, and the 6.3 as side b. The unknown side will be side c. Now, the corresponding opposite angles are 33 for angle A, x for angle B, and an unknown angle C. Since we don't have any information for either angle C or side c, we can skip that part of the law of sines. We are left with this: 4.7/sin(33) = 6.3/sin(x). Solving this for our variable x, we get this: x = 46.89.

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