What is Ambiguous Case of the Law of Sines?

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In trigonometry, the law of sines is a formula that develops a relationship between a triangle's angles and the lengths of its sides. As long as you know at least two sides and one angle, or two angles and one side, you can use the law of sines to find the other missing pieces of information about your triangle. However, in a very limited set of circumstances you can end up with two answers to the measure of one angle. This is known as the ambiguous case of the law of sines.

The law of sines is especially useful for any non-right triangle because it adjusts the traditional sine ratio to work for many different triangles. The law of cosines is another tool that uses a pythagorean type relationship to determine the sides of a triangle.

When the Ambiguous Case Can Happen

The ambiguous case of the law of sines can only happen if the "known information" part of your triangle consists of two sides and an angle, where the angle is ‌not‌ between the two known sides. This is sometimes abbreviated as an SSA or side-side-angle triangle. If the angle were between the two known sides, it would be abbreviated as an SAS or side-angle-side triangle, and the ambiguous case would not apply. Other examples of triangle formations include SSS (which is a unique triangle), ASA (also a unique arrangement), and AAA which is not unique.

Given such a triangle where only SSA is known, the possible number of triangles is no longer limited. We can still use the sine rule with the law of sines to solve for a given side or opposite angles, but there are two possible outcomes.

SSA Triangle Configuration
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The second triangle created here by ‘swiveling’ side a further in creates an oblique triangle. Side ‌b‌, side ‌a‌, and angle ‌A‌ are held constant, but angle ‌C‌, angle ‌B‌ and side ‌c‌ are variable depending on the other given information.

A Recap of the Law of Sines

The law of sines can be written two ways, given triangle ABC where ‌a,b,c‌ are side lengths of the triangle and ‌A,B,C‌ are the corresponding angles across from each side. The first form is convenient for finding the measures of missing sides:

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

The second form is convenient for finding the measures of missing angles:

\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}

Note that both forms are equivalent. Using one form or the other won't change the outcome of your calculations. It just makes them easier to work with depending on the solution you're looking for.

How to Use the Law of Sines

In most cases, the only clue that you might have an ambiguous case on your hands is the presence of an SSA triangle where you're asked to find one of the missing angles . Imagine you have a triangle with angle ‌A‌ = 35 degrees, side ‌a‌ = 25 units and ‌b‌ = 38 units, and you've been asked to find the measurement of angle ‌B‌. Once you find the missing angle, you must check to see if the ambiguous case applies.

Insert your known information into the law of sines. Using the second form, this gives you:

\frac{\sin 35}{25} = \frac{\sin B}{38} = \frac{\sin C}{c}

We can disregard the third term with ‌C‌ and ‌c‌ as we are only trying to solve for angle B (which can then give us the remaining information) so it's irrelevant for the purposes of this calculation. So really, you have:

\frac{\sin 35}{25} = \frac{\sin B}{38}

Solve for ‌B‌. One option is to multiply the denominator across; this gives you:

38 \times \frac{\sin 35}{25} = \sin B

Next, simplify by using a calculator or chart to find the value of sin(35). It's approximately 0.57358, which gives you:

\sin B = \frac{38}{25} \times 0.57358

This then simplifies through, giving you:

\sin B = 0.8718416

To finish solving for ‌B,‌ take the arcsin or inverse sine of 0.8718416. Or, in other words, use your calculator or chart to find the approximate value of an angle B that has the sine 0.8718416. That angle is approximately 61 degrees.

Checking for the Ambiguous Case

Now that you have an initial solution, it's time to check for the ambiguous case. This case pops up because for each acute angle, there is an obtuse angle with the same sine. So while ~61 degrees is the acute angle that has a sine value equivalent 0.8718416, you must also consider the obtuse angle as a possible solution. This is a little tricky because your calculator and your chart of sine values most likely won't tell you about the obtuse angle, so you have to remember to check for it.

Find the obtuse angle with the same sine by subtracting the angle you found – 61 degrees – from 180 (known as its supplementary angle). So you have 180 - 61 = 119. So 119 degrees is the obtuse angle that has the same sine as 61 degrees. (You can check this with a calculator or sine chart.)

But will that obtuse angle make a valid triangle with the other information you have? You can easily check by adding that new, obtuse angle to the given angle from the original problem. If the total is less than 180 degrees, the obtuse angle represents a valid solution, and you'll have to continue any further calculations with ‌both‌ possible triangles in consideration. If the total is more than 180 degrees, the obtuse angle does not represent a valid solution.

In this case, the given angle was 35 degrees, and the newly discovered obtuse angle was 119 degrees. As 119 + 35 = 154 degrees which is less than 180, the ambiguous case applies and you have two valid solutions: the angle in question can measure 61 degrees, or it can measure 119 degrees. It is important to note that the third unknown side also varies with these angle measures, and so does the non-included angle.

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