In this video, we’re going to look at the special relationships that exist between
the trigonometric ratios for complementary angles. So I’m starting off with a right-angled triangle in which I’ve labelled the three
sides as 𝑥, 𝑦, and 𝑧. I haven’t specified any units here because it doesn’t particularly matter,
but we’re assuming they’re all the same, so all centimetres or inches or whatever. I’ve also
labelled the two angles that aren’t right angles as 𝛼 and 𝛽. Now 𝛼 and 𝛽 are the complementary angles that we’re interested in, because
remember complementary angles, the definition, is that they sum to ninety degrees. And of course as this is a triangle, the total sum of its angle is a hundred and
eighty degrees as the other angle is a right angle, ninety degrees. That means that there’s
ninety degrees left over for 𝛼 and 𝛽, and therefore, their sum must be ninety.
So 𝛼 and 𝛽 are the complementary angles that we’re interested in. So what I’m going to do is I’m just gonna work with each of these angles in turn.
I’m going to start with angle 𝛼. And first I’m gonna label the three sides of this triangle in
relation to angle 𝛼, so the hypotenuse, the opposite, and the adjacent. The hypotenuse of a right-angled triangle is of course always the same side, but
remember then the adjacent and the opposite change depending on which angle you’re interested
in. So in relation to angle 𝛼, 𝑦 is the opposite and 𝑥 is the adjacent.
Now what I’m gonna do is I’m gonna write down the three trigonometric ratios
sine, cosine, and tangent for angle 𝛼. So sine, first of all, sine remember is opposite divided by hypotenuse so that
will be 𝑦 divided by 𝑧. Cosine is adjacent divided by hypotenuse so that will be 𝑥 divided by 𝑧. Finally tangent is opposite divided by adjacent so that will be 𝑦 divided by 𝑥. If you struggle to recall any of these definitions of sine, cosine, or tangent,
then you need to remember that word SOHCAHTOA in order to help you with that. So SOHCAHTOA, remember, takes the first letter of each of these words. So in the
case of sine, for example, sine is opposite divided by hypotenuse and that’s where the SOH comes
Right. Now I’m gonna do exactly the same thing, but with angle 𝛽. So I’m gonna
label the sides again, but in relation to angle 𝛽 this time. The hypotenuse, as we said, is
always the hypotenuse. But this time for angle 𝛽, the opposite is this side here, side 𝑥. And the
adjacent is this side here, side 𝑦. And now just like I did for angle 𝛼, I’m gonna write down the three
trigonometric ratios for angle 𝛽 in terms of 𝑥, 𝑦, and 𝑧. So for 𝛽, sine of 𝛽 is opposite over hypotenuse. So looking at the orange labels
this time, that will be 𝑥 divided by 𝑧. For cosine, cosine is adjacent divided by hypotenuse. So again looking at the
orange labels, that will be 𝑦 divided by 𝑧. Finally tangent is opposite divided by adjacent. So looking at the orange labels,
that will be 𝑥 divided by 𝑦.
So I have the three trigonometric ratios, sine, cosine, and tangent written down
for both 𝛼 and 𝛽, this pair of complementary angles, in terms of 𝑥, 𝑦, and 𝑧. Now if you just have a look at these different trigonometric ratios, there are a
couple of things that will become apparent. If you look at sine of 𝛼, that is equal to 𝑦 over 𝑧. And what you’ll notice on the right-hand side of the screen that we also have 𝑦
over 𝑧 again, but this time it’s equal to cos or cosine of 𝛽. So what this tells us then is that sine of angle 𝛼 must be equal to cosine of
angle 𝛽. So what we have then is that sine of 𝛼 is equal to cos of 𝛽. Now remember that 𝛼 and 𝛽 are complementary angles, so we had on top of the
screen that 𝛼 plus 𝛽 is equal to ninety degrees. We could also write this down alternatively
as 𝛽 is equal to ninety minus 𝛼. And so if I replace 𝛽 with this expression in terms of 𝛼, then it will give me
a general relationship. So I have this relationship here which is that sine of any angle 𝛼 is equal to
cosine of ninety minus that angle. That relationship will always hold true for complementary
Now looking back at these ratios, you’ll also see that there are two places where
𝑥 over 𝑧 appear. So cos of 𝛼 is equal to 𝑥 over 𝑧 and sine of 𝛽 is also equal to 𝑥 over 𝑧. So therefore, I must have that cos of 𝛼 is equal to sine of 𝛽. Again, if I replace 𝛽 with ninety minus 𝛼, then it gives me another general
relationship for complementary angles. So I have that cos of 𝛼 is equal to sine of ninety minus 𝛼.
Finally if you look at tangent, we have tan of 𝛼 is 𝑦 over 𝑥 and tan of 𝛽 is 𝑥
over 𝑦. So in fact, these two tangents are reciprocals of each other, because the fractions are
just inverted in each case. So the relationship here is that tan of 𝛼 is equal to one over tan of 𝛽. And again, if I then write this relationship down where I replaced 𝛽 with ninety
minus 𝛼, then I have the general relationship that tan of 𝛼 is equal to one over tan of
ninety minus 𝛼. So where you have a pair of complementary angles, that is angles that sum to
ninety degrees, these three relationships will exist between the values of their trigonometric
ratios, so between the sine, cosine, and tangents. And that’s due to the fact that the opposite
and the adjacent sides in the right-angled triangle swap round for the two complementary
Okay let’s just see how to use what we’ve just worked out in order to apply it to a
question. So we’re given a table of information in which were given the values of the sine,
cosine, and tangent ratios for two angles: twenty degrees and sixty degrees. And we’re asked to
use this information to write down cos of thirty and sine of seventy. Now of course, we’re
assuming in this question that we haven’t got access to a calculator, so we just need to use
the information in the table and nothing else. So thinking about what we just did and complementary angles, well what was spot
as that we’re asked something about thirty degrees. But we’re given some information about
sixty degrees. And they’re complementary angles, because they sum to ninety, the same thing with
the seventy degrees and twenty degrees.
So let’s think about cos of thirty first of all. We saw in the previous page that
cos of an angle 𝛼 is equal to sine of ninety minus that angle. So I can use this to write down something about cosine of thirty and sine of
ninety minus thirty. So I have the cos of thirty is equal to sine of ninety minus thirty. Now of
course, ninety minus thirty is just sixty. So I have that cos of thirty is equal to sine of sixty. And using the information
in the table, I can look that value up and see that it’s equal to root three over two. That’s an
exact value as a surd. So I’ve just used the relationship that exists between sine and cosine for this
pair of complementary angles and then information in the table in order to work that out.
Indeed, if you do have a calculator, you can then confirm that this is in fact the correct value.
Okay, the method for the second one is very similar then. So sine of seventy degrees,
well remember sine of 𝛼 is equal to cos of ninety minus 𝛼. So we can use a very similar process. Sine of seventy will be cos of ninety minus
seventy or cos of twenty. And then if I look this value up in the table, cos of twenty to three significant
figures is zero point nine four zero. So in summary then, we demonstrated the relationship that exists between the
trigonometric ratios for complementary angles and then we’ve seen the type of question where
you may need to apply this knowledge.