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 90 degrees. And of course, as this is a triangle, the total sum of its angles is 180 degrees. As the other angle is a right angle, 90 degrees, that means that there’s 90 degrees left over for 𝛼 and 𝛽. And therefore, their sum must be 90. 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 from.
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, sin 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 sin 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 sin of angle 𝛼 must be equal to cos of angle 𝛽. So, what we have then is that sin 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 90 degrees. We could also write this down alternatively as 𝛽 is equal to 90 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 sin of any angle 𝛼 is equal to cos of 90 minus that angle. That relationship will always hold true for complementary angles.
Now, looking back at these ratios, you’ll also see that there are two places where 𝑥 over 𝑧 appears. So, cos of 𝛼 is equal to 𝑥 over 𝑧 and sin of 𝛽 is also equal to 𝑥 over 𝑧. So, therefore, I must have that cos of 𝛼 is equal to sin of 𝛽. Again, if I replace 𝛽 with 90 minus 𝛼, then it gives me another general relationship for complementary angles. So, I have that cos of 𝛼 is equal to sin of 90 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 replace 𝛽 with 90 minus 𝛼, then I have the general relationship that tan of 𝛼 is equal to one over tan of 90 minus 𝛼.
So, where you have a pair of complementary angles, that is, angles that sum to 90 degrees, these three relationships will exist between the values of their trigonometric ratios, so between their 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 angles. 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 we’re given the values of the sine, cosine, and tangent ratios for two angles, 20 degrees and 60 degrees. And we’re asked to use this information to write down cos of 30 and sin of 70. 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 is that we’re asked something about 30 degrees. But we’re given some information about 60 degrees. And they’re complementary angles, because they sum to 90, the same thing with the 70 degrees and 20 degrees.
So, let’s think about cos of 30 first of all. We saw in the previous page that cos of an angle 𝛼 is equal to sin of 90 minus that angle. So, I can use this to write down something about cos of 30 and sin of 90 minus 30. So, I have that cos of 30 is equal to sin of 90 minus 30. Now, of course, 90 minus 30 is just 60. So, I have that cos of 30 is equal to sin of 60. 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 it- this is in fact the correct value. Okay, the method for the second one is very similar then. So, sin of 70 degrees, well remember sin of 𝛼 is equal to cos of 90 minus 𝛼. So, we can use a very similar process. Sin of 70 will be cos of 90 minus 70 or cos of 20. And then, if I look this value up in the table, cos of 20, to three significant figures, is 0.940.
So, in summary then, we’ve 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.