### Video Transcript

In this video, our topic is the
sine rule. This is a rule that applies to all
triangles, and it allows us to solve for interior angles as well as side
lengths. In this lesson, we’ll learn what
this rule says, as well as how to use it practically.

As we get started talking about the
sine rule, also sometimes called the law of sines, we can recall that this word
“sine” describes a mathematical function. And specifically, if we take values
of 𝑥 ranging from zero up to two 𝜋 and insert them into this function, then out
comes a series of values ranging from a maximum value of one to a minimum value of
negative one.

An important point here is that
these 𝑥-values, the argument of the sine function, are angles. We see here that we’ve written out
these angles in units of radians, so they range from zero up to two 𝜋 radians. But we could equivalently write
them in degrees. 𝜋 over two radians is 90 degrees,
𝜋 radians is 180 degrees, and so on, up to two 𝜋 radians, which is 360
degrees. So the sine function is a
trigonometric function that takes as its argument an angle expressed either in
radians or degrees. And it returns a value as high as
positive one and as low as negative one.

The sine rule then uses this
function, and it uses it specifically with triangles. And one great thing about the sine
rule is that it applies to virtually any triangle. It could be a right triangle; an
isosceles triangle, where two of the angles are identical; or just generally any
three-sided shape. What the sine rule does
mathematically is it equates ratios of sines of an angle to corresponding side
lengths.

Say we have a triangle, and the
interior angles of this triangle are called 𝐴, 𝐵, and 𝐶 like this. When we talk about the sine rule
equating ratios of sines of angles to corresponding side lengths, what we mean is
that each one of these angles, say 𝐵, for example, has what we can call a
corresponding side length of the triangle. And that in the case of 𝐵 is this
length right here. For angle 𝐶, that side length
would be here. And then for angle 𝐴, it’s the
side length opposite that, right here.

What we can do then is name the
side lengths of this triangle to correspond with the interior angles they stand
opposite to. So we’ll call this side length here
lowercase 𝑎. And then the side length opposite
angle 𝐶 we’ll call lowercase 𝑐. And that opposite 𝐵 we’ll give
lowercase 𝑏 as its name. And just to be clear, the symbol
for this angle here is meant to be an uppercase 𝐶.

Our triangle now is completely
labeled, all the angles and all the side lengths. And we can go back to our statement
about the sine rule, that it describes ratios of sines of an angle to corresponding
side lengths. In any triangle, there are always
three angles and three side lengths, and so we’ll get three ratios.

The first ratio, we can say, is the
sin of the angle capital 𝐴 divided by the side length lowercase 𝑎. The second ratio is the same thing
but now for the angle and side we’ve marked 𝐵 and then lastly for the angle and
side marked 𝐶. Now that we have these ratios, the
last step is to equate them. So we can go and write that the sin
of angle 𝐴 divided by side length 𝑎 equals the sin of angle 𝐵 divided by side
length 𝑏, which is likewise equal to the sin of angle 𝐶 divided by side length
𝑐.

This overall expression here is the
mathematical statement of the sine rule. What it helps us do is solve for
interior angles and side lengths in triangles in whatever application we find
them. We’ll apply this rule using a few
examples in a moment. But before we do, note that while
this is perhaps the most common form for the sine rule, there’s another
mathematically equivalent way to write it.

If we were to invert all these
ratios so that now the side lengths 𝑎, 𝑏, and 𝑐 in lowercase were on top and the
sines of all the angles are in the denominators, this is still a mathematically
correct way of writing out the sine rule. One way to convince ourselves of
this is to start here and then use cross multiplication until we have our expression
in this form. With that said, the best way to
learn the sine rule is to use it in practice. So let’s now look at an example
exercise.

What is the length of side 𝑎 of
the triangle shown?

Looking at this triangle, we see
that two of the interior angles are given to us. Along with that, the side length
corresponding to one of these interior angles is also known. We also see side length 𝑎, which
is opposite the angle marked out as 47 degrees. And it’s this length we want to
solve for. To do this, we’re going to use the
sine rule, also known as the law of sines. This rule says that if we have a
triangle, and it could be any triangle, then the ratio of the sine of any one of the
triangle’s interior angles to its corresponding side length is equal to that same
ratio for any of the other angles and corresponding sides.

Now, we don’t need to know all
three of the angles or all three of the sides to use any one of these relations. Because of this, we can apply the
sine rule to our triangle over here in order to solve for side length 𝑎. We see that the side length, as we
noted, is opposite the interior angle of 47 degrees. So then, as one of our ratios
applying the sine rule, we can write the sin of 47 degrees divided by 𝑎. And then by that same rule, this
ratio is equal to the sine of any other interior angle in our triangle divided by
its corresponding side length. The question is, do we have that
information?

Well, we see that we have this
other interior angle, 95 degrees, and that opposite that angle we indeed have a
known side length, 10 centimeters. So our ratio then is the sin of 95
degrees divided by 10 centimeters. And as we said, the sine rule tells
us this is equal to the sin of 47 degrees divided by 𝑎.

And now that we have this equation,
all we need to do is rearrange algebraically to solve for 𝑎. If we multiply both sides by 𝑎,
that factor cancels out on the left. And then, next, we multiply both
sides of the equation by 10 centimeters divided by the sin of 95 degrees, canceling
out 10 centimeters and that sine on the right. And finally, we have an expression
that we can evaluate to solve for side length 𝑎. Entering this expression on our
calculator, to two significant figures, we find a result of 7.3 centimeters. That’s the length of side 𝑎 in the
triangle in our diagram.

Let’s look now at a second
example.

What is the size of angle 𝐴, in
degrees, in the triangle shown?

In this triangle, we see the angle
𝐴 marked out here, as well as the interior angle of 54 degrees. Opposite this 54-degree angle is a
side length of 8.4 centimeters. And then opposite angle 𝐴 is a
side length of 9.6 centimeters. We see then that this triangle is
well set up for us to use the sine rule to solve for this angle 𝐴. The sine rule applies to any
triangle. It says that if the interior angles
and the corresponding side lengths are marked out as shown, then the sine of any of
those interior angles divided by the corresponding side length is equal to that same
ratio for any of the other pairs of angles and sides.

So as we think about applying the
sine rule to our triangle over here, specifically to solve for angle 𝐴, we remind
ourselves that angle 𝐴 corresponds to the side length of 9.6 centimeters and the
angle of 54 degrees corresponds to 8.4 centimeters. So then the ratio of the sin of the
unknown angle 𝐴 to 9.6 centimeters is equal, by the sine rule, to the sin of 54
degrees divided by 8.4 centimeters.

And now what we want to do is to
isolate this angle 𝐴. To do this, we’ll first multiply
both sides of the equation by 9.6 centimeters, canceling that factor on the left and
then giving us this expression here. Notice that on the right-hand side
of our equation, these units of centimeters cancel from numerator and
denominator.

And now what we need to do is to
invert or undo the application of the sine function on the angle 𝐴 we want to solve
for. We do this by applying what’s
called the arc sine or the inverse sine to the sin of 𝐴. And then to maintain our equality,
we apply the same inverse sine function to the right-hand side of our
expression. When we take the inverse sine of
the sin of the angle 𝐴, what remains is simply the angle 𝐴.

Our final step is to evaluate the
right-hand side of this expression. And just as the sine function is
standard on any scientific calculator, so the inverse sine function will be as
well. When we evaluate this expression
and find our answer in degrees, to two significant figures, it’s 68 degrees. That’s the size of the angle 𝐴 in
the triangle shown.

Let’s now consider one last example
exercise.

What is the length of side 𝑎 of
the triangle shown in the diagram?

In this diagram, we see side 𝑎
right here, as well as two interior angles of this triangle, 64 degrees and 38
degrees. And then opposite this angle of 38
degrees, we have a side length given as 6.1 centimeters. Knowing this, we want to solve for
the side length 𝑎. And we’re going to do it by
applying what’s called the sine rule. This rule says that if we have a
triangle, and it could be any triangle, with its angles and sides marked out like
this, then the ratio of the sine of any of these angles to the corresponding side
length is equal to that same ratio for the other pairs of sides and angles.

We can see that a key to being able
to use the sine rule is to have at least two corresponding angle and side pairs. That way, we can set up an equality
and then solve for any one of those four values, given the other three. As we look to apply the sine rule
to our diagram in order to solve for the side length 𝑎, we see that we have one
such angle and side pair, 38 degrees with 6.1 centimeters, but that that’s the only
one. We don’t have a second
corresponding pair. If we knew what this angle of the
triangle was though, we see that that angle is opposite the side length 𝑎 we want
to solve for.

It turns out that we can solve for
this angle, but we won’t do it using the sine rule. Instead, we’ll use the fact that
the sum of the interior angles of any triangle is always 180 degrees. So therefore, if we call this
unknown angle in our triangle capital 𝐴, then we can say that 𝐴 plus 64 degrees
plus 38 degrees is equal to 180 degrees. Now, 64 degrees plus 38 degrees is
102 degrees. And if we subtract this amount from
the left-hand side, the negative 102 cancels with positive 102. And we find that 𝐴 equals 180
degrees minus 102 degrees, or 78 degrees.

Now that we know this angle in our
triangle, we can use the sine rule to solve for the side length 𝑎. We’ll write that the sin of capital
𝐴, which we know is 78 degrees, divided by the side length lowercase 𝑎 is equal to
this ratio over here, the sine of this angle in our triangle divided by this side
length. Solving for 𝑎 is now just a matter
of cross multiplying. We can start by multiplying both
sides of the equation by 𝑎, leading that factor to cancel on the left. And then, in our resulting
equation, if we want to isolate 𝑎 on the right-hand side, we do this by multiplying
both sides by the inverse of this ratio. When we multiply this ratio by its
inverse, that product is equal to one, effectively canceling all this out. And finally, we have an expression
we can enter in on our calculator to solve for 𝑎. To two significant figures, 𝑎 is
9.7 centimeters. That’s the length of this side of
the triangle.

Let’s now summarize what we’ve
learned about the sine rule. In this lesson, we learned that the
sine rule applies to all triangles. And it equates ratios of sines of
angles to corresponding side lengths. This means that given a triangle,
with its angles and sides labeled like this, the sine rule tells us that the sin of
angle 𝐴 divided by side length 𝑎 is equal to the sin of angle 𝐵 divided by side
length 𝑏, which is also equal to the sin of angle 𝐶 divided by side length 𝑐. Lastly, we saw that an equivalent
expression for the sine rule is to put the side lengths on top in the numerators and
the sines of the angles in denominators. Either way of stating the rule is
correct and can be used to solve for side lengths and interior angles of
triangles.