Video: The Law of Cosines

Video Transcript

In this video we’re gonna to look at the law of cosines. We’re then gonna look at how can be applied to non-right-angled triangles to find out the length of missing sides or angles. So let’s start off with a non-right-angled triangle 𝐴𝐵𝐶.

Now we labelled the vertices capital 𝐴, capital 𝐵, and capital 𝐶, and we label the opposite side to those vertices with lowercase 𝑎, lowercase 𝑏, and lowercase 𝑐. Now there are two different ways of arranging the law of cosines: one’s best if you’re trying to find a missing side, and one’s best if you’re trying to find a missing angle.

And here’s the arrangement that’s best for finding the length of a missing side. Now it tells us that the length of this side squared is equal to the length of this side squared plus the length of this side squared minus two times the length of this side times the length of this side times the cosine of this angle here.

So when we know the length of two adjacent sides and the measure of the angle between them, the enclosed angle, then we can easily work out the length of the opposite side to that angle. And the other version of the law of cosines helps us to find the measure of an angle when we know the length of all three sides.

Now this second version here is just a direct rearrangement of this version here. If I add two 𝑏𝑐 cos 𝐴 to both sides, I’ve got 𝑎 squared plus two 𝑏𝑐 cos 𝐴 on the left and negative two 𝑏𝑐 cos 𝐴 plus two 𝑏𝑐 cos 𝐴 makes nothing, so I’m just left with 𝑏 squared plus 𝑐 squared on the right-hand side.

Now if I take away 𝑎 squared from both sides, that leaves me with two 𝑏𝑐 cos 𝐴 is equal to 𝑏 squared plus 𝑐 squared minus 𝑎 squared. Then dividing both sides by two 𝑏𝑐 gives us this formula here. So you either need to memorize both of these versions or you need to be able to remember one of them and know how to rearrange it into the other format.

The first version then is used to calculate the length of a third side when you know the length of the other two sides and the angle in between them, and the second version is used to find the measure of an angle when you know the length of all three sides. Now let’s apply these techniques to a few different problems.

𝐴𝐵𝐶 is a triangle where 𝑎 equals 12 centimeters, 𝑏 equals 20 centimeters, and 𝑐 equals 26 centimeters. Find the measure of the smallest angle in triangle 𝐴𝐵𝐶 giving your answer to the nearest second.

Now in this question, we’re given the lengths of all three sides and we’re asked to find the measure of one of the angles, so this is the perfect situation to use the law of cosines. Now I’d recommend with all these questions that you do a quick sketch of the problem, doesn’t have to be 100 percent accurate, but it needs to represent the problem; it just makes it easier to visualise what we’re going to do.

So here’s our sketch and remember that this is side 𝑎, opposite angle 𝐴, this is side 𝑏, opposite angle 𝐵, and this is side 𝑐, opposite angle 𝐶. And we’ve been asked to find the measure of the smallest angle. And in triangles, the smallest angle is always opposite the smallest side, and the largest angle is always opposite the largest side, so we’re looking for angle 𝐴. The smallest side is side 𝑎, 12 centimeters, so the smallest angle is gonna be angle 𝐴.

So in order to find the measure of this angle here, we’re going to use this version of the formula: cosine of angle 𝐴 is equal to 𝑏 squared plus 𝑐 squared minus 𝑎 squared all over two 𝑏𝑐. Now let’s take the information from the diagram and substitute the values for 𝑎, 𝑏, and 𝑐 that we already know. And this tells us that the cosine of angle 𝐴 is equal to 20 squared plus 26 squared minus 12 squared all over two times 20 times 26.

And with a bit of work on our calculator, that simplifies to 233 over 260. And if cosine of angle 𝐴 is equal to 233 over 260, then angle 𝐴 is equal to the inverse cosine of 233 over 260. And that turns out to be 26.342975 etc. etc. degrees. But the question asked us to give our answer to the nearest second, so we’ve gotta convert from decimal degrees into degrees, minutes, and seconds.

So I’ve got 26 whole degrees plus 0.342975 and so on degrees. And remember, a minute is one sixtieth of a degree. So if I take this decimal and multiply it by 60, we can see that it represents 20.5785 and so on minutes. And that tells us we’ve got 20 whole minutes plus 0.5785 and so on minutes. And remember that there are 60 seconds in a minute. So if I multiply that 0.5785 and so on by 60, and when I do that, 0.5785 and so on minutes is 34.711 and so on seconds.

So my angle is 26 degrees 20 minutes and 34.711 and so on seconds. But I was asked to give my answer to the nearest second, so I need to round 34.711 and so on to the nearest whole number. So the answer is that the measure of the smallest angle in this triangle is 26 degrees, 20 minutes, and 35 seconds to the nearest second.

So just to summarize our approach there, we were given all three side lengths in this question, and we had to find the measure of an angle, so we knew that this version of the law of cosines formula was gonna help us to answer the question. We did a quick diagram, and then it was just a matter of substituting in the relevant values for 𝑎, 𝑏, and 𝑐 and evaluating that in order to give our final answer.

Now 𝐴𝐵𝐶 is a triangle where 𝐵𝐶 is equal to 25 centimeters, 𝐴𝐶 is 13 centimeters, and the measure of angle 𝐶 is 142 degrees. Find the length 𝐴𝐵 giving your answer to three decimal places.

As we said before, it’s always good to do a quick sketch diagram in order to help you gather your thoughts on what you’ve got to do. And when we look at this, we’ve been given the length of this side, the length of this side, and the size of the measure of their enclosed angle here. And we’ve been asked to find the length of this side over here. Now this situation is a classic case for using this version of the law of cosines, or cosine formula. But wait! We’re trying to work out the length of side 𝑐, and we were given the length of side 𝑎 and the length of side 𝑏 and the measure of angle 𝐶.

The letters don’t quite match. We’re gonna have to look at the pattern in the formula. 𝑏 and 𝑐 are the adjacent sides, and 𝐴 is the enclosed angle. But in our problem, 𝑎 and 𝑏 are the adjacent sides, and 𝐶 is the enclosed angle. So we can replace them correspondingly in our formula: 𝑎 squared plus 𝑏 squared, the two adjacent sides, minus two times 𝑎 times 𝑏 times cos 𝐶, the enclosed angle, is equal to the square of the side that we’re looking for.

Now we can substitute in the values that we’ve got in the question. So that means length 𝐴𝐵 all squared is equal to 25 squared plus 13 squared minus two times 25 times 13 times cos 142. And just before we go on, I’ll just draw your attention to this term here. We’re taking away two times 25 times 13 times cos 142. Now the multiplication signs between all of those mean that they go together. I would strongly recommend putting parentheses or brackets around those terms just to make sure that everybody knows that they’re all one term. And when I put that into my calculator, I’ve got that length 𝐴𝐵 squared is equal to 1306.20699 and so on. But of course I don’t want the length 𝐴𝐵 squared, I just want the length 𝐴𝐵. So I need to take square roots of both sides, and that tells me that length 𝐴𝐵 is 36.141485 and so on.

Now looking back at the question, it wants us to give our answer to three decimal places, so I need to do some rounding. And not forgetting units, all the measurements were in centimeters that we had, so the answer 𝐴𝐵 is 36.141 centimeters to three decimal places of accuracy.

Now just to summarize our approach here, this pattern here where we were given two sides, two adjacent sides of the triangle, we were given those lengths and the measure of the enclosed angle, and we needed to find the length of the side opposite that. That told us it was the law of cosines. We had to rearrange that a bit, and then we could just substitute in the values that we were given in the question and then round to the appropriate level of accuracy at the end.

Now 𝐴𝐵𝐶 is a triangle where 𝑎 equals 18 centimeters, 𝑏 equals 10 centimeters, and the measure of angle 𝐶 is 76 degrees. Find the measure of angle 𝐴 giving your answer to one decimal place.

Well again let’s start off by doing a quick sketch diagram. So that’s triangle 𝐴𝐵𝐶 with side 𝑎 is equal to 18 centimeters, side 𝑏 is equal to 10 centimeters, and the measure of angle 𝐶 is 76 degrees. And we’ve been asked to find the measure of angle 𝐴 to one decimal place. When we think about this, we’ve been given the length of two adjacent sides and the measure of the enclosed angle, which is screaming at us law of cosines.

But what that would enable us to do is to work out the length of side 𝑐 down here. That’s not the measure of angle 𝐴. But if we knew the length of side 𝑐, we’d know the length of all of the sides of this triangle and we’d know the measure of one of the angles, so we could then use the law of sines to work out the measure of angle 𝐴.

So that’s what we’re gonna do. Now recall, the law of cosines in this particular format tell us that 𝑎 squared is equal to 𝑏 squared plus 𝑐 squared minus two 𝑏𝑐 cos 𝐴. But that’s when 𝑏 and 𝑐 are the adjacent sides and 𝐴 is the enclosed angle. But we’ve got sides 𝑎 and 𝑏 as the adjacent sides and 𝐶 as the enclosed angle.

So substituting in that pattern, we’ve got side 𝑐 squared is equal to side 𝑎 squared plus side 𝑏 squared minus two times the length of side 𝑎 times the length of side 𝑏 times the cosine of angle 𝐶. And we can simply substitute in the values we’ve got for 𝑎 and 𝑏 and angle 𝐶, and we can see that side 𝑐 squared is equal to 18 squared plus 10 squared minus two times 18 times 10 times the cosine of angle 76 degrees.

And remember it’s a good idea to always bracket those together. And when we tap that all into our calculator, we find that 𝑐 squared is equal to 336.90811 and so on. Then taking square roots of both sides tells us that the length 𝑐 is 18.355057 and so on centimetres. So let’s make a note of that back on our diagram, and then we can see that we’re looking for angle 𝐴, and we know the side opposite of that is 18 centimeters. We know side 𝑐, and we know angle 𝐶, so this is telling us that the law of sines will be useful.

Now hopefully you can remember the two ways of writing out the law of sines: side length 𝑎 over sine of angle 𝐴 equals side length 𝑏 over sine of angle 𝐵 equals side length 𝑐 over sine of angle 𝐶 or sine of angle 𝐴 over side length 𝑎 equals sine of angle 𝐵 over side length 𝑏 equals sine of Angle 𝐶 over side length 𝑐. Now we’re trying to find angle 𝐴; we know side length 𝑎; and we know angle 𝐶; and we know side length 𝑐.

So given that we’re looking for the measure of angle 𝐴, it’s better that that’s on the numerator. So this is the version of the law of sines that we’re going to use. And in fact, we’re just gonna pick out these two terms. So having written that down, we can now just substitute in the values that we know from the question. And that tells us that sine of angle 𝐴 over 18 is equal to sine of 76 degrees over 18.355057 and so on.

Now we’re trying to solve to find the value of 𝐴, so if I multiply both sides of my equation by 18, then the 18s are going to cancel from the left-hand side. And that means that sine of angle 𝐴 is equal to 18 times sine of 76 degrees over 18.355057 and so on. Now if sine of angle 𝐴 gives us that ratio, inverse sine of that ratio will give us the measure of angle 𝐴.

Now hopefully you kept that 18.355057 and so on on your calculator, and you can now use that so you get a nice accurate answer for this value here. And when you do the tapping in on your calculator, you get 𝐴 is equal to 72.08734532 and so on degrees. But of course, we only want our answer to one decimal place, so we need to round to one decimal place. And that means that angle 𝐴 is 72.1 degrees to one decimal place of accuracy.

So in this question, we started off with this pattern of two adjacent side lengths and an enclosed angle, which enabled us to work out the length of another side using the law of cosines. And we had to follow that up with the pattern that we saw here; we wanted to know an angle, but we knew the opposite side length, and then we knew a side and the angle opposite it, so we were able to use the law of sines to finish off the question.

Now in exams and on longer questions, this need to use a combination of the law of cosines and the law of sines is actually quite common. Alright then, let’s have a quick final summary before we go.

With the law of cosines then, if we know the lengths of two adjacent sides and an enclosed angle, say sides 𝑏 and 𝑐 and angle 𝐴, then we can work out the length of the other side opposite the angle that we know. Or if we know the lengths of all three sides, then we can work out the measure of one of the angles. So those patterns give rise to these two different formulae. And if we know different sides labelled in a different way, we’re able to use those patterns to rearrange those letters accordingly.