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Video: Trigonometric Ratios for Complementary Angles

Lauren McNaughten

Use the nonright angles in a right triangle to identify the relationships that exist between the trigonometric ratios for complementary angles and apply this knowledge to solve problems without using a calculator.

09:08

Video Transcript

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 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, 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 angles.

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 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 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.