### Video Transcript

In this video, we’re going to look
at the ambiguous case of the law of sines. Suppose you’re given some
information about a triangle such as the lengths of two of its sides and the size of
one of its angles. The ambiguous case occurs when that
information doesn’t define a unique triangle. But in fact, it’s possible to draw
more than one triangle using the information given. More specifically, it occurs when
the angle that you’re given is not included between the two sides. So let’s look at an example then to
clarify what we mean.

I have here a triangle in which I
have the lengths of two sides, five centimetres and six centimetres, and the size of
one angle. And you’ll see that that angle is
not included between the two sides. So it’s not between the sides of
six centimetres and five centimetres. I’m asked to calculate the size of
angle 𝐴. So I’m gonna try to answer this
problem then using the law of sines. So I need to recall what it tells
me. Remember then that it tells me that
the ratio between each side and the sine of its opposite angle is equal. Now I’ve chosen to write it in this
format here where the sines of the angles are in the numerator because this
particular question asked me to calculate an angle, so it will be slightly more
straightforward to use this version than it will be to use the version where the
sides are in the numerator.

So I’m going to write down then
this law of sines using the information in this question. So I will have that sine of angle
𝐴 over six is equal to sine of 50 over five. This gives me an equation that I
can solve in order to work out 𝐴. So the first step is to multiply
both sides of this equation by six. And in doing so, I have that sine
of 𝐴 is equal to six sin 50 over five. Now to work out angle 𝐴, I need to
do sine inverse of both sides. So I have that 𝐴 is equal to sin
inverse of six sin 50 over five. Now I’m gonna use my calculator to
evaluate this. And when I do, it tells me that 𝐴
is equal to 66.817 or 67 degrees to the nearest degree.

So, brilliant! Fantastic! We’ve finished the question you
might think. But have a look at the diagram and
have a look specifically at angle 𝐴. Now this diagram isn’t supposed to
be deliberately misleading. And what you notice is that angle
𝐴 is an obtuse angle. So there’s no way that angle 𝐴 can
be equal to 67 degrees. It has to be somewhere between 90
and 180 degrees. So what’s gone wrong? Well, nothing in the maths that
we’ve done so far is incorrect. All of those steps that we followed
are perfectly fine. It’s just that those steps haven’t
actually found us angle 𝐴 in this particular triangle because there’s another
triangle we can draw that would use this same information.

If I extend the base of the
triangle out in this direction here, and if I were then to take my compass and put
it on point 𝐵 and set it to five centimetres and then draw a little arc of points,
you’d see that actually it crosses this line again. And actually, there’s another point
on this baseline here that is still five centimetres away from 𝐵. And I’ll label that then as 𝐴
dash. So I have two triangles: I have
triangle 𝐴𝐵𝐶, the original triangle, and I have triangle 𝐴 dash 𝐵𝐶. And both of those triangles have
the same information in them. So in both of those two triangles,
it’s true that the measure of angle 𝐶 is 50 degrees, 𝐵𝐶 is six centimetres, and
either 𝐴𝐵 or 𝐴 dash 𝐵 — depending on which triangle you’re talking about — is
equal to five centimetres. So this collection of information
doesn’t uniquely specify a single triangle.

What all of this means then is that
the angle we’ve just calculated, 67 degrees, isn’t actually angle 𝐴. It’s angle 𝐴 dash which we can see
is an acute angle. So we have this angle here. However, we can use this to work
out angle 𝐴. If you look at the triangle formed
by 𝐴, 𝐵, and 𝐴 dash, you’ll see that that’s an isosceles triangle because it has
those two sides of the same length, both five centimetres. What that means then is that the
other base angle must also be equal to 67 degrees. And then, we can work out angle 𝐴
that we’re looking for by doing 180 minus 67 as you’ll see those two angles sit on a
straight line together. So subtracting 67 from 180 gives us
a value of 113 degrees. And this is then the size of the
angle 𝐴 that we were looking for.

This is illustrative of a general
property of the sine ratio, which is that for an angle 𝜃 between zero and 180
degrees, sine of 𝜃 is in fact equal to sine of 180 minus 𝜃. This relationship always holds true
for supplementary angles. So in this question, we applied the
law of sines correctly and we got an answer for angle 𝐴. But then we saw that our answer
couldn’t actually be the correct value given the context of the diagram and the fact
that angle 𝐴 is an obtuse angle. If we hadn’t been given the diagram
to refer to and instead we’ve just been given the list of information in green, then
either these two values for 𝐴 would be valid. So 𝐴 could either be 67 degrees or
113 degrees. And we’d have to provide two
possible answers to this problem.

This problem tells us 𝐴𝐵𝐶 is a
triangle where the measure of angle 𝐵 is 110 degrees, side 𝑏 is 16 centimetres,
and 𝑐, side 𝑐, is 12 centimetres. How many possible solutions are
there for the other lengths and angles?

Now the inclusion of the word
“possible” here tells me that there’re a couple of situations that could exist. It could be that the information
I’ve got doesn’t actually describe a triangle at all and it’s not possible to draw a
triangle that fulfills those requirements. Or it could be that that
information describes a unique triangle, in which case there’s only one possible
solution for the other lengths and angles. It could also be that this is an
example of the ambiguous case of the law of sines and in fact there are two possible
solutions for the other lengths and angles.

So I’m gonna start off with a
sketch of what triangle 𝐴𝐵𝐶 could look like. Now as I said, it may be the case
that it doesn’t exist at all or it may be the case that there’s more than one
triangle I could draw. But I’ll start off with just one
idea. So triangle 𝐴𝐵𝐶 could look like
this. And I’ve put all the relevant
information onto my diagram. Now I’m gonna start off by trying
to calculate angle 𝐶. And I’m going to use the law of
sines for that. Remember this is what the law of
sines tells me, that the ratio between the sine of each angle and the opposite side
is constant throughout the triangle. So I’m going to use information I
know, which is side 𝑏 and angle 𝐵 and also side 𝑐. And it’s angle 𝐶 I’m looking to
calculate.

So substituting this information
gives me then that sine of 110 over 16 is equal to sin 𝐶 over 12. Now I’m looking to solve this
equation to work out angle 𝐶. So I’m gonna multiply both sides by
12. And I’ve just swapped the order of
the two sides around here. But it tells me that sine of 𝐶 is
equal to 12 sin 110 over 16. In order to work out angle 𝐶 then,
I need to use the inverse sine function. And this tells me that angle 𝐶 is
equal to sine inverse of 12 sin 110 over 16. Now I’ll use my calculator to
evaluate this. And doing so, I get that 𝐶 is
equal to 44.8109 or 45 degrees, when rounded to the nearest degree.

So this tells me then that there is
at least one possible solution for the other lengths and angles. I’ve worked out angle 𝐶 is 45
degrees. So I could work out angle 𝐴 by
subtracting the 45 degrees and the 110 degrees from 180, which is the angle sum of
the triangle. And then consequently, I could
apply the law of sines again to work out the size of side 𝑎. So now the question is are there
actually two possible solutions. Is this an example of the ambiguous
case of the sine rule when there’s another possible value for 𝐶 and hence another
possible value for 𝐴 and so on?

So if you recall the way that we
worked out the other possible value of the angle is we subtract this from 180. And if I were to do that, I would
get that 𝐶 is equal to 135 degrees. Now the question is, is this
possible? Is it okay for 𝐶 to be equal to
135 degrees? And to determine whether that’s the
case, we need to look back at the original triangle. We already knew angle 𝐵 in the
triangle was 110 degrees. This means then that it can’t
possibly be the case that angle 𝐶 is 135 degrees because if I add those two
together, it would exceed 180 degrees, which is the angle sum for a triangle. What this means then is that angle
𝐶 must be that 45 degrees that I originally worked out. And therefore, there is a unique
solution to this problem. So there’s only one possible
solution for the lengths of the sides and the sizes of the angles. So this check is essential. If you think there is a second
possible solution, you must look at the other angles in the triangle and confirm
that it doesn’t exceed the angle sum.

In summary then, we’ve seen what
the ambiguous case of the law of sines is. We’ve seen how and why to
rises. And we’ve seen how to determine a
second possible solution if one exists.