Lesson video

Hi, I'm Mr. Bond.

And in this lesson, we're going to learn when to use the sine and cosine rules.

By now, we should know how to use the sine and cosine rules.

If this is the first time that you're coming across the sine and cosine rules, then you might want to go back to these lessons before coming back to this one.

In this lesson, we're going to focus on when to use the sine and cosine rules.

Let's start by looking at the sine rule.

This is often how it's written down.

But when we use the sine rule, we only use two of the expressions at a time.

So let's think about what we need to know.

In order to use the sine rule.

We either need to know two angles and one side, or two sides and the non included angle.

So if we know either two angles and a side, or two sides and a non included angle, we'll use the sine rule.

Here's the cosine rule.

What do we need to know in order to be able to use the cosine rule? Well, we could use it if we knew three sides.

Or we could also use it if we knew two sides and the included angle.

So these are the two conditions in which we could use the cosine rule.

Here's an example.

We're not actually going to calculate anything here.

I just want to know should the sine rule or the cosine rule be used to find the value of x? Well, we need to start by thinking about what we know.

We know three sides.

So we just learned on the previous slide, that we should use the cosine rule.

What about this triangle? Should the sine rule or cosine rule be used to find the value of x? Again, we need to start by thinking about what we know.

We know two angles and a side.

So that means that we should use the sine rule.

Here are some questions for you to try.

For each diagram, decide whether the sine or cosine rule is needed to find the length or angle labelled x.

Pause the video to complete the task and resume the video when you're finished.

Here are the answers.

Let's work from left to right.

So for the first triangle, we knew three sides, so we'd use the cosine rule.

For the next triangle along, we know two sides and a non included angle.

So we'd use the sine rule.

For the third triangle along, we know two sides, and the included angle.

So that's the cosine rule.

And then for our final triangle, we know two angles and a side.

So that's the sine rule.

In this example, a student is working out the length of the side label w, they're working out is below in the colour shown here.

I'd like you to do two things.

Check they're working out and correct it if necessary.

And also say, would you have worked it out in this way? And explain why.

Pause the video to have a think and resume the video when you're finished.

Well first, let's check they're working and correct it if necessary.

So let's look at what they've done in the first part of their working here.

They've used the sine rule to work out a side that they've called x.

That must be the side that they've labelled as six metres.

If you check this in your calculator, you'll see that x is very close to being equal to six metres, and they've clearly just rounded.

They've then use the cosine rule, because they know that they have two sides and an included angle.

So they've worked out w to be equal to 5.

63.

And again, if you check their calculations on a calculator, you'll see that it's correct.

Just a couple of things then.

They've said that x was equal to six metres, but they've rounded.

So what does that mean for their value w? It wouldn't be precisely accurate.

Also, have a look at the information we were initially given, two angles and a side.

That means that we could have just used the sine rule to work out the length w straight away, without needing to use both rules.

Here's our final example to think about today.

We know two angles and a side for this triangle.

Which rule does that tell us we should use? It tells us that we should use the sine rule.

Let's have a little bit more of a think about that.

Let's label our sides a, b, and c.

And we'll write out the full sine rule.

Let's substitute in what we know.

We know that length a is four, angle B is 45 degrees, and angle C is 110 degrees.

We also know that the length c is x, that's what we're trying to find.

So given that the thing that we're trying to find appears in this part of the rule, we need to say that's equal to either sine a over four, or sine 45 over b.

So what if we put it equal to sine a over four? We have two values that are known, but two variables.

So we can't put these two equal to each other and solve for x.

What about if we put sine 110 over x equal to sine 45 over b? Well, once again, we have two known values, but two unknown variables.

So we can't put these equal to each other and solve either.

So how do we know that we can use the sine rule? If we know two angles and a side? It doesn't seem to have worked here.

Well, if we know two angles in a triangle, and we know that angles in a triangle sum to 180 degrees, we can always work out the third angle.

So that means that we could work out the value of angle A, and then put sine A over four equal to sine 110 over x and solve for x.

That's all for this lesson.

Thanks for watching.