Video Transcript
In this video, we are going to look
at the trigonometric ratios: sine, cosine, and tangent. Which are the names given to the
ratios that exist between the different pairs of sides in right-angled triangle. So, let’s begin by looking at the
right-angled triangle on the screen here. And to start off with, we need to
be familiar with the names that are used for the sides in a right-angled
triangle.
Now you should already be familiar
with the name hypotenuse, which is the name given to the longest side of a
right-angled triangle. The side which is opposite the
right angle. And you may perhaps have met that
before in the Pythagorean theorem. The other two sides of a
right-angled triangle also have particular names. And their names are in relation to
the angle that we’re interested in. So in this right-angled triangle
here, I’ve labeled one of the other two angles, the ones that aren’t right
angles. I’ve labeled one of them as 𝜃. So the names of the sides are in
relation to this angle 𝜃. The first label is the
opposite. And just like the hypotenuse is
opposite the right angle, the opposite side is the one opposite this angle 𝜃. So that would be this side
here. The name that we give to the final
side, the third side, is the adjacent. And that side is adjacent or next
to both this angle 𝜃 and the right angle. It’s between those two. That’s how we recognize which of
the sides is the adjacent. So you need to be familiar with
these three labels: the opposite, the adjacent, and the hypotenuse. And you need to be comfortable
identifying which side is which in various different right-angled triangles.
So the three trigonometric ratios
that we’re interested in are the ratios between the different pairs of the sides of
the right-angled triangle. The first one that we’re going to
look at has the name sine. Now for a particular angle 𝜃, this
sine ratio is the ratio between the opposite and the hypotenuse. So its definition is that sin of
the angle 𝜃 is equal to the opposite divided by the hypotenuse. Whatever the lengths of those two
sides are. The second ratio is given the name
cosine or often abbreviated to cos. And this is the ratio between the
adjacent and the hypotenuse. So it is defined as the cos of the
angle 𝜃 is equal to the adjacent divided by the hypotenuse. The final ratio is called tangent,
often abbreviated to tan. And this is the ratio between the
opposite and the adjacent sides. So its definition is that tan of
the angle 𝜃 is equal to opposite divided by adjacent. So that’s how these three ratios
are defined.
And what’s important is that these
ratios, for a particular angle 𝜃, they are fixed no matter how big I choose to draw
the triangle. And therefore, no matter what the
lengths of the opposite, the adjacent, and the hypotenuse are. For a particular angle 𝜃, the
values of these three ratios will always be the same. So if I were to extend this
triangle, but still keeping that angle 𝜃 constant. Then the value of this sine ratio
would be the same, whether I use the opposite and the hypotenuse as they’re labeled
in that solid triangle. Or whether I use the opposite and
the hypotenuse as they are now marked in red in the larger triangle. In both cases, when I do the
opposite divided by the hypotenuse, I will get the same value for this ratio. And of course, the same thing is
true for cosine and for tangent as well.
Now to help you remember the
definitions of sine, cosine, and tangent and which sides are involved, there is a
little acronym that you can use to help. And so what we do is we take the
first letter of each of those words. So sin 𝜃 is equal to opposite over
hypotenuse. So that gives us SOH, S, O, H. Cos is equal to the adjacent over
hypotenuse. That gives us CAH, and so on. So SOHCAHTOA. If you remember that, then you’ll
be able to remember the definitions of sine, cosine, and tangent more easily. If you wanted to convince yourself
that these ratios are in fact constant for a fixed angle 𝜃, you could do your own
investigation. So like the one I’ve started on the
screen, you could draw out a triangle, measure these sides as accurately as
possible, and calculate the ratios. And then continue to a medium
triangle and a larger triangle and convince yourself that the ratios do remain the
same. So let’s look at how we can use
this to answer some questions.
The first question, we’re given
a diagram of a right-angled triangle. And we’re asked to write down
the value of cos 𝜃.
So for me, when I’m answering
any problem to do with trigonometry, the first step for me is always to label
the three sides of the triangle in the problem with their labels. So the opposite, the adjacent,
and the hypotenuse. So I’ve just used the first
letter of those words here. Remember the hypotenuse is
opposite the right angle. The opposite is opposite the
angle 𝜃. And the adjacent is between 𝜃
and the right angle. Now we’re asked about cos of
𝜃. So we need to recall the
definition of cos of 𝜃. And if you remember SOHCAHTOA,
well CAH is adjacent and hypotenuse. So the definition of cos 𝜃 is
that it’s the adjacent divided by the hypotenuse. What I need to do then is just
write down what this ratio is for this particular triangle. So I need to replace the
adjacent and the hypotenuse with their values in this example.
So looking at the triangle
then, the adjacent is 12 centimeters. So I’m gonna have cos 𝜃 is
equal to 12 over — and the hypotenuse. Looking at the diagram, that’s
13. So I have that cosine of 𝜃,
cos 𝜃, is equal to 12 over 13. So all I needed to do in this
question was recall the definition of cos and then write down the ratio using
the values that are specific to this question.
Okay, our second question. We’re given another
right-angled triangle. And the angle that we’re
interested in is labeled 𝛼 this time rather than 𝜃. And the question asked us to
write down the value of sin of 𝛼.
So step one, as in the previous
question. First of all, I’m just gonna
label each of these three sides as the opposite, the adjacent, and the
hypotenuse. Next, I need to recall what the
definition of sine is. So SOH, sine is opposite
divided by hypotenuse. So looking at the diagram, I
can see that I know the hypotenuse. It’s 10 millimeters. But I don’t actually know what
the opposite is. That length hasn’t been given
to me in the diagram. But I need to know in order to
write down this sine ratio. So there’s a little bit of work
that I need to recall in order to work this out. And it’s the Pythagorean
theorem. Because remember, the
Pythagorean theorem tells us about the relationship that exists between the
three sides in a right-angled triangle. And what the Pythagorean
theorem enables me to do is calculate the length of the third side, if I know
both of the other two. And in this case I do. I can see they’re 10
millimeters and six millimeters.
Now the Pythagorean theorem,
you often see it written as 𝑎 squared plus 𝑏 squared is equal to 𝑐
squared. But remember, what the theorem
actually tells us is that if you take the two shorter sides of a right-angled
triangle. So that’s 𝑎 and 𝑏. And if you square them and add
them together, then it gives you the same result as if you square the longest
side, the hypotenuse. So I can use the Pythagorean
theorem to work out the length of this third side. Now I’m not gonna call it 𝑂
because that might be confused with zero. So I’m gonna give it a
different letter. Let’s call it 𝑏. So in my working for this part
of the question, I’m gonna think of that third side as 𝑏. So what I’m gonna do is I’m
gonna write out the Pythagorean theorem. But I’m gonna replace 𝑎 with
six and keep 𝑏 as it is. And I’ll replace 𝑐 with
10. So I have six squared plus 𝑏
squared is equal to 10 squared.
Next, I’m gonna write down what
six squared and 10 squared are. So I’ve 36 plus 𝑏 squared is
equal to 100. Now I want to solve this
equation for 𝑏. So I’m gonna subtract 36 from
both sides. So I have 𝑏 squared is equal
to 64. And then in order to work out
𝑏, I need to square root both sides. So I’ll have 𝑏 is equal to the
square root of 64, which is eight. So this tells me then that the
length of that third side, which is the opposite, must be eight millimeters. Now you may actually have been
able to spot that without going through the formal working out because six,
eight, 10 is an example of the Pythagorean triple. That is a right-angled
triangle, where the lengths of all three sides are integers. And if you knew that and
recognized it, then you could cut down on a little bit of the working out
here.
Right, the reason we wanted to
know the length of this third side is because we wanted to write down the sine
ratio. So remember, sine is opposite
divided by hypotenuse. And now I know the opposite is
eight millimeters. And the hypotenuse is 10
millimeters. So this tells me then that sin
of 𝛼 must be eight over 10. Now that could be simplified to
four-fifths. Or, we could write it as a
decimal. So we have our answer to this
question, which is that sin of 𝛼 is equal to 0.8.
Okay, the final example in this
video then.
We’re asked to write down the
value of tan 𝜃 in this right-angled triangle here.
Now, we’re given the lengths of
all three sides in this particular triangle. We’re not going to need
them. So first step as always, we
need to label the three sides: the hypotenuse, the opposite, and the
adjacent. Next, we need to recall what
the definition of that tan ratio is. So TOA, tan is the opposite
divided by the adjacent. So looking at our diagram,
we’re gonna be using the opposite which is root two over two and the adjacent
which is a half. So because both of these are
fractions, I don’t want to write one over the other and end up with a fraction
that has four layers within it. So I’m gonna write tan 𝜃 is
root two over two divided by a half, to start off with.
Now I need to simplify
that. And there’s a couple of
different ways you can think about this. Dividing by a half is the same
as multiplying by two. Or, you can think about when
you divide by a fraction, you invert it. You flip it over. And you multiply instead. Both of those ways of thinking
about this would lead me to the same point which is that tan of 𝜃 is equal to
root two over two multiplied by two. Now this two in the numerator
and this two in the denominator are gonna cancel each other out. And therefore, I’m left with an
answer that tan 𝜃 is equal to root two. So I’m gonna leave my answer
like that in surd form because of course, that’s exact at the moment. Whereas if I tried to evaluate
it as a decimal, I’d need to do some rounding.
So to summarize then, we’ve defined
the three trigonometric ratios: sine, cosine, and tangent as the ratios between
different pairs of sides in a right-angled triangle. We’ve seen how to recognize the
difference between sine, cosine, and tangent by using that acronym SOHCAHTOA. And we’ve seen how to write down
the value of each of these trigonometric ratios from a diagram of a right-angled
triangle with different lengths labelled.
