, Ambiguous case of the law of sines. Explained in a video Tutorial with pictures and many practice problems
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Ambiguous case of the Law of sines

Explained with examples

Video Tutorial on the Ambiguous Case

First, something very important to review

Well, the ambiguous case goes back to something you learned in your prior trig lessons and it has to do with a characteristic of the sine function. So, first let's look at something that is always true about the sine function.
What do you notice about the $$ sin (30 ^{\circ}) $$ vs $$ sin (150 ^{\circ}) $$

Quadrant I sine

Quadrant II sine

Both $$sin(30)$$ and $$sin(150)$$ have the exact same measure !

This is a big deal

This is a big deal! And it is the foundation for the ambiguous case of the law of sines. (Remember ambiguous means that something has more than 1 meaning).

As you can see, two different angles have the same sine value !

So, if I asked you :

What angle measurement has a sine value of $$\frac {1}{2} ? $$

You could say two different things--either $$ 30^{\circ}$$ or $$ 150^{\circ} $$ . This is what we mean by ambiguous.

Maybe, you are already getting the idea here. If not, no problem. This is much easier to understand by looking at a specific example .

A Specific Example

The best way to see this ambiguity is to solve a problem using the Law of Sines like normal... and, at the end..you'll notice something.


Use the law of sines to find the measure of $$\angle \blue c$$ in triangle 1 below. Round your answer to the nearest whole number.

Quadrant I Triangle


$$ \frac{sin(\blue c)}{20} = \frac{sin(29)}{11} \\ sin( \blue c ) = \frac{20 \cdot sin(29)}{11} \\ sin( \blue c ) = \red{ 0.881472 } \\ m\angle \blue c = sin^{-1}( \red{ 0.881472 } ) \\ $$

$ \text{ Guess what?} \\ \blue 2 \text { different angle measurements} \\ \text{ can be used !} \\ \boxed{ \blue c = 62^{\circ} \text { or } \blue c = 118^{\circ} } \\ \text{ In other words, both} \\ sin( \blue { 62} ) = \red{ 0.881472 } \\ \text{ and } \\ sin( \blue{118} ) = \red{ 0.881472 } $
So there are actually, 2 completely different triangles that we can make up because 2 different angles have a sine value of 0.881472 .

Quadrant I Triangle

Quadrant I Triangle
This occurs because the sine function has both a quadrant I and a quadrant II angle that have the exact same sine value!
Quadrant I Triangle
Quadrant II Triangle

Can we always get 2 triangles?

Answer:

In a word ,no

No, sometimes only one triangle is possible. In some cases, zero triangles are possible!

Taken from our free downloadable worksheet

$$ \frac{ sin( \red M)}{15} = \frac{sin(55)}{27} \\ sin( \red M)= \frac{15 \cdot sin(55)}{27} \\ \angle \red M = 27 ^{\circ} \\ $$

$$ \text{What about Quadrant II? } \\ \text{In Quadrant II, we have 153 } \\ \angle \red M = 180-27 = 153 ^{\circ} $$

$ \text{ You can }\red{not} \text{ use } 153^{\circ} \\ $

Can you figure out why the Quadrant II angle of $$153^{\circ}$$ can not work?

Because $$\angle L$$ is $$ 55^{\circ} $$ and the total degrees of the angles inside a triangle have to be $$ 180 ^{\circ}$$ .

Take away: The Quadrant II angle will sometimes be too large and in cases like this, there is only 1 solution and only 1 possible triangle.

Only 1 Triangle

The Rule

The Rule When you add up the given angle ($$55 ^{\circ}$$) and the Quadrant II angle ( $$153^{\circ}$$), the sum must be less than $$180^{\circ} $$ .

This is because you still have a third angle $$\angle N$$ in Triangle 3 below.

Third Angle for Rule

A common type of question is to ask how many triangles are possible. As you can see, sometimes there are 2 triangles, sometimes (like the last examples), only one traingle is posssible.

If you're curious about whether or not no triangles are possible, then check this out , but first I recomend you make sure you can do the practice problems below

Steps for Solving Problems

Question: For Triangle 4 below, how many triangles can be formed?

Triangle 4

Taken from our free downloadable worksheet

Step 1
Use the law of sines to solve for an angle

The only angle we can solve for is $$\angle \red c$$

$$ \frac{sin( \red c)}{24} = \frac{sin(16)}{17} \\ sin( \red c)= \frac{24 \cdot sin(16)}{17} \\ \red c= sin^{-1} (\frac{24 \cdot sin(16)}{17}) \\ m\angle \red c \approx 22^{\circ} $$

Note: we found the Quadrant I angle.

Step 2

Calculate the Quadrant II angle.

$$ \boxed{\text{ Quadrant II}} \\ \red c = 180 - 22^{\circ} \\ m\angle \red c = 158^{\circ} $$

Step 3

Check to see if the Quadrant II angle is a possible angle measurement for the triangle, using the sum of the angles of a triangle .

$$ 158^{\circ} + 16^{\circ} < 180^{\circ} $$

Answer: In this case, there are 2 different triangles possible .

Some Things to remember. When your calculator computes the inverse sine($$sin^{-1}$$) to give you an angle, the calculator always provides the acute angle from Quadrant I. So, you always have to manually calculate the Quadrant II angle, and then check to see if that angle is possible.

Practice Problems

Problem 1

For $$\triangle ABC$$, find all possible values of $$m\angle B $$ to the nearest tenth of a degree.

Practice problem 1

Step 1
Use the law of sines to solve for $$\angle B$$

$$ \frac{sin(\red B)}{53}= \frac{sin(54^{\circ})}{62} \\ sin(\red B)= \frac{53 \cdot sin(54^{\circ})}{62} \\ \red B= sin^{-1} \left( \frac{53 \cdot sin(54^{\circ})}{62} \right) \\ \red B= 43.7 ^{\circ} $$

Step 2

Calculate the Quadrant II angle.

$$ \boxed{\text{ Quadrant II}} \\ \red B = 180 - 43.7^{\circ} \\ m\angle \red B = 136.3^{\circ} $$

Step 3

Check to see if the Quadrant II angle is a possible angle measurement for the triangle, using the sum of the angles of a triangle.

$$ 136.3^{\circ} + 54^{\circ} < 180^{\circ} $$

Answer: In this case, there is only 1 possible $$m\angle B$$ , because there is only 1 possible triangle.

To keep practicing similar problems, just scroll to the next one. If you're curious about whether or not, it's possible for there to be zero triangles, then read our article here.

Problem 2

For $$\triangle DEF$$, $$ d=6, e =24, \text{ and } m\angle E=38^{\circ}$$. How many Triangles can be formed?

Step 1
Use the law of sines to solve for $$\angle D$$ (The only angle we can directly solve for.)

$$ \frac{sin(\red D)}{d}= \frac{sin( E) }{e} \\ \frac{sin(\red D)}{6}= \frac{sin(38^{\circ})}{24} \\ sin(\red D)= \frac{6 \cdot sin(38^{\circ})}{24} \\ \red D = sin^{-1} \left( \frac{6 \cdot sin(38^{\circ})}{24} \right) \\ \red D = 8.9 ^{\circ} $$

Step 2

Calculate the Quadrant II angle.

$$ \boxed{\text{ Quadrant II}} \\ \red D = 180 - 8.9^{\circ} \\ m\angle \red D = 171.1^{\circ} $$

Step 3

Check to see if the Quadrant II angle is a possible angle measurement for the triangle, using the sum of the angles of a triangle

$$ 171.1^{\circ} + 38^{\circ} < 180^{\circ} $$

Answer: In this case, there is only 1 possible

Problem 3

For $$\triangle LMN $$, $$ l=30, m=24, \text{ and } m\angle M=40^{\circ}$$. Find all possible $$m\angle L$$ to the nearest degree.

Step 1
Use the law of sines to solve for $$\angle D$$ (The only angle we can directly solve for.)

$$ \frac{sin(\red L)}{l}= \frac{sin( M) }{m} \\ \frac{sin(\red L)}{30}= \frac{sin(40^{\circ})}{24} \\ sin(\red L)= \frac{30 \cdot sin(40^{\circ})}{24} \\ \red L = sin^{-1} \left( \frac{30 \cdot sin(40^{\circ})}{24} \right) \\ \red L = 53.5 ^{\circ} $$

Step 2

Calculate the Quadrant II angle.

$$ \boxed{\text{ Quadrant II}} \\ \red L = 180 - 53.5^{\circ} \\ m\angle \red L = 126.5^{\circ} $$

Step 3

Check to see if the Quadrant II angle is a possible angle measurement for the triangle, using the sum of the angles of a triangle

$$ 126.5^{\circ} + 40^{\circ} < 180^{\circ} $$

Answer: In this case, there are 2 possible $$m\angle L$$ : $$53.5^{\circ} \text{ and } 126.5^{\circ} $$ .

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