Video Transcript
In this video, our topic is the
cosine rule. This rule, sometimes also called
the law of cosines, uses the cosine function to let us solve for sides and angles in
triangles. In this lesson, we’ll learn this
rule, and we’ll also get some good practice using it. As we get to know this rule, let’s
recall some information about the cosine function.
Say that we have a right triangle
like this one and we also know the lengths of all three sides of the triangle. The hypotenuse we’ll call ℎ, the
side length opposite this angle we’ve defined, 𝛼, we’ll call 𝑜, and then the side
length adjacent to that angle 𝛼 we’ll call 𝑎. If we were to solve for the cos of
this angle 𝛼 — and by the way, the argument in a cosine function does always need
to be an angle, whether in degrees or radians — then on our right triangle, the
cosine of this angle is equal to the adjacent side length 𝑎 divided by the
hypotenuse ℎ. So, this is how we define the
cosine of an angle that’s part of a right triangle.
And in general, we can see how the
cosine function behaves by plotting it against a variety of input angles. If we write out those angles in
units of radians and we let them range from zero up to two 𝜋, then over this range,
the cosine function looks like this. It reaches a maximum value of
positive one at zero and two 𝜋 radians. And then it goes all the way down
to negative one at an angle of 𝜋 radians. And just as a side note, we could
also write these angles in degrees. So, for example, 𝜋 over two
radians is 90 degrees, or three 𝜋 over two radians is 270 degrees.
Knowing this about the cosine
function, let’s see how this function works within the cosine rule. We mentioned earlier that we use
the cosine rule to solve for information in triangles. And any kind of triangle is a
candidate for this rule. This includes right triangles like
we saw over here in defining the cosine function, isosceles triangles which have two
equal angles and therefore two equal sides, as well as triangles that don’t fit
either of these two descriptions but are just general three-sided objects.
The way the rule is typically
presented, we imagine a triangle with interior angles labeled 𝐴, 𝐵, and 𝐶, all
capital letters. And then, we name the corresponding
sides of the triangle opposite these angles with similar letters but in
lowercase. So for example, side 𝑎 would be
written like this, side 𝑏 like this, and side 𝑐 like this. We understand then that each one of
our capital letters refers to an angle either in degrees or radians, and that each
one of our lowercase letters corresponds to a side length in some distance unit,
whether centimeters or meters or whatever it may be.
With this set up, the way the
cosine rule works is we pick a corresponding angle and side pair. In general, that could be any one
of our three pairs: angle 𝐵 and side 𝑏, angle 𝐶 and side 𝑐, and so on. But the conventional way to present
the cosine rule is to choose angle 𝐴 and side 𝑎. Once we’ve made that selection, we
then choose the remaining two side lengths in our triangle. Those would be lowercase 𝑏 and
lowercase 𝑐. Using these four variables,
lowercase 𝑎, 𝑏, and 𝑐 and uppercase 𝐴, we can write out the mathematical form of
the cosine rule. This rule says that the side length
𝑎 squared is equal to the side length 𝑏 squared plus the side length 𝑐 squared
minus two times 𝑏 times 𝑐 all multiplied by the cos of this angle, capital 𝐴.
In this equation then, we see this
pair of angle and corresponding side, upper and lowercase 𝑎, as well as the
remaining two side lengths, lowercase 𝑏 and 𝑐. The two things this equation leaves
out, then, are the other two interior angles, capital 𝐵 and capital 𝐶. It’s important to see, though, that
this particular form of the cosine rule, which is a common one and one we may see if
we look it up in a textbook, doesn’t put a limit on which particular angle or side
length we can solve for. In this particular triangle we’ve
drawn here, we could have, for example, made this angle here, angle 𝐴, or this
angle here. It’s an arbitrary choice that we
chose this particular angle to give that label.
The basic idea of this rule then is
that given this set of four variables, the three side lengths of our triangle and
one of the interior angles, if we know three of these values, then we can use the
cosine rule to solve for the fourth. And typically, this involves
solving either for side length 𝑎 or the angle capital 𝐴. As we do that, one important thing
to keep in mind is that this term here on the left-hand side is 𝑎 squared. It can be easy to forget that if we
want to solve for 𝑎 itself, we’ll need to take the square root of both sides. Keeping that in mind, let’s get
some practice now using the cosine rule through an example.
Find the length of side 𝑎 of the
triangle shown.
Okay, so in this triangle, we have
an interior angle marked out 49 degrees, and opposite that is the side length we
want to solve for, 𝑎. To help us do this, we’re given the
other two side lengths: 9.4 centimeters and 12 centimeters. And knowing all this about this
triangle, we can use what’s called the cosine rule to solve for the side length
𝑎. This rule says that if we have any
triangle and we label the interior angles capital 𝐴, 𝐵, and 𝐶 and that if we call
the corresponding side lengths lowercase 𝑎, 𝑏, and 𝑐, then the side length 𝑎
squared is equal to the side length 𝑏 squared plus the side length 𝑐 squared minus
two times 𝑏 times 𝑐 all multiplied by the cos of the angle 𝐴.
So, this rule is well set up for
letting us solve for the unknown side length 𝑎 in our triangle. To solve for that side length,
we’ll want to figure out which information in this triangle corresponds to side
length 𝑏 and to side length 𝑐 and to angle 𝐴. We can see right away that it’s
this 49-degree angle that we’ll call angle 𝐴 for our cosine rule equation. But then, what about our two side
lengths? Which one of these two is 𝑏 and
which one will we call 𝑐? As we look at our cosine rule
equation though, we can see that this choice doesn’t make a mathematical
difference. Whichever side length we call 𝑏
and calling the other one 𝑐, we’ll find the same answer for our side length 𝑎.
Just to make a particular choice
though, let’s call our side length of 12 centimeters 𝑏. And that means our side length of
9.4 centimeters is 𝑐. Now that we know the side lengths
𝑐 and 𝑏 and the angle 𝐴, we have all the information we need to fill in the
right-hand side of this equation. So, we can write then that our 𝑎
squared, where 𝑎 is this unknown side length we want to solve for, is equal to 12
centimeters squared plus 9.4 centimeters squared minus two times 12 centimeters
times 9.4 centimeters times the cos of 49 degrees.
Before we enter this expression on
our calculator, there’s one last step to take. We want to solve for 𝑎 rather than
𝑎 squared. So let’s take the square root of
both sides of this equation. When we do that, on the left-hand
side, the power of two cancels out with our square root, which is effectively a
power of one-half. And now we have an expression for
the side length 𝑎 we want to solve for. When we calculate the right-hand
side, to two significant figures, we find a result of 9.2 centimeters. This is the length of side 𝑎 in
our triangle.
Let’s look now at a second example
exercise.
Find the size of angle 𝐴, in
degrees, of the triangle shown.
Looking at this triangle, we see
that all three side lengths are given to us, and we want to solve for this unknown
angle here. Given this information, we can use
the cosine rule to solve for this angle 𝐴. The cosine rule tells us that given
a triangle with interior angles capital 𝐴, 𝐵, and 𝐶 and corresponding side
lengths lowercase 𝑎, 𝑏, and 𝑐, then if we pick out the particular side length
we’ve labeled 𝑎, that length squared is equal to the side length 𝑏 squared plus
the side length 𝑐 squared minus two times 𝑏 times 𝑐 all multiplied by the cos of
the angle 𝐴.
As we look to apply the cosine rule
to our scenario though, we can see it’s not any side length we want to solve for but
rather an interior angle. We can say then that in the cosine
rule equation, it’s this angle here that we want to solve for. And we can do that by rearranging
this expression. As a first step to doing that, we
can subtract 𝑏 squared and 𝑐 squared from both sides. This means that on the right-hand
side, we have positive 𝑏 squared minus 𝑏 squared canceling that out and positive
𝑐 squared minus 𝑐 squared canceling that out as well.
Next, we’ll continue to move toward
isolating this angle 𝐴 by dividing both sides of the equation by negative two times
𝑏 times 𝑐, which means on the right-hand side, this factor of negative two times
𝑏 times 𝑐 cancels. That leaves us with this
expression. And the last thing we’ll do to get
this angle 𝐴 all by itself is to invert our operation of the cosine function by
taking the inverse cosine or arc cosine of both sides. On the right-hand side, when we
take the inverse cos of the cos of the angle 𝐴, those inverse cosine and cosine
functions effectively cancel out. So now, we have an expression
letting us solve for the angle 𝐴.
Notice that to do this, we need to
know all three of the side lengths of our triangle, lowercase 𝑎, 𝑏, and 𝑐. And as we look over at the
particular triangle on this example, we see we do have that information. We can say that 7.8 centimeters is
our side length lowercase 𝑎. And although it doesn’t make a
mathematical difference whether we call this side length 𝑏 or this one 𝑏, which
would mean that the remaining side length we call side length 𝑐, for the sake of
clearly identifying our values, let’s call this 14-centimeter side length side
length 𝑏. And that means 9.6 centimeters is
𝑐.
And now we can insert these side
lengths into an equation to solve for the angle 𝐴. It’s equal to the inverse cos of
this whole argument where we’ve plugged in values for side lengths 𝑎, 𝑏, and 𝑐,
respectively. Notice, by the way, that all our
side lengths have the same units, centimeters. Since they’re all consistent on
that basis, we can go ahead and calculate the right-hand side of this equation. Rounding our answer to two
significant figures, we find a result of 32 degrees. This is the size of the angle 𝐴 in
this triangle.
Let’s now look at one last example
exercise.
Find the length of side 𝑎 of the
triangle shown.
In our triangle, we see this side
length opposite an angle of 103 degrees. Along with this, based on these
markings on the other two sides of the triangle, we can tell that they have the same
length. So this side length here is 7.1
centimeters also. As a side note, this means we’re
working with a special kind of triangle called an isosceles triangle. But anyway, our mission is to solve
for this side length here, and we can do it using a rule called the cosine rule,
also sometimes called the law of cosines.
This rule tells us that given a
triangle with interior angles marked out capital 𝐴, 𝐵, and 𝐶 and corresponding
side lengths lowercase 𝑎 and 𝑏 and 𝑐, we can solve for the square of one of the
sides by adding together the square of the other two sides and then subtracting from
that two times those other two sides multiplied by the cos of an angle we’ve called
capital 𝐴, where in our triangle, this angle capital 𝐴 is opposite the side length
lowercase 𝑎 that we’re solving for. So in our actual triangle over
here, to solve for this side length, we’ll apply the cosine rule.
To do this, we’ll need to identify
what the other two side lengths, they’re called 𝑏 and 𝑐 in this equation, and the
angle opposite the side length we want to solve for, called capital 𝐴 here,
are. Looking at our triangle, we see
that angle opposite the side length we’re solving for is 103 degrees. So that’s our angle capital 𝐴. And then, as far as identifying 𝑏
and 𝑐, these values in our cosine rule equation, we see that the other two side
lengths in our triangle are both the same. In this case then, both 𝑏 and 𝑐
are 7.1 centimeters. This is the case because, as we saw
earlier, we’re working with an isosceles triangle.
Knowing all this, we can now plug
in these values into their place on the right-hand side of our cosine rule
expression. And once we’ve done this, we’re
very close to being able to solve for the unknown side length 𝑎. The one thing we don’t want to
forget, though, is that right now we have an expression for 𝑎 squared. To solve just for 𝑎, we’ll want to
take the square root of both sides, where on the left, this square root and the
square term will cancel one another out. And now, when we enter this
expression on the right-hand side into a calculator, rounding to two significant
figures, we find a result of 11 centimeters. This is the length of side 𝑎 in
our triangle.
Let’s summarize now what we’ve
learned about the cosine rule. In this lesson, we saw that the
cosine rule helps us solve for unknown angles and side lengths in any triangle. Given a general triangle with
interior angles marked out as capital 𝐴, 𝐵, and 𝐶 and corresponding side lengths
of lowercase 𝑎 and 𝑏 and 𝑐, this rule tells us that the length of one of the
sides of this triangle, we’ve called it 𝑎, squared is equal to the sum of the
square of the other two side lengths minus two times the product of those two
remaining sides all multiplied by the cos of this angle capital 𝐴, where this angle
is opposite the side length lowercase 𝑎 that we’re solving for. This is a summary of the cosine
rule.
