Lesson video

Hello, I'm Mrs. Lashley, and I'm gonna be talking you through the lesson today.

I really hope you're willing to try your best and even if it gets challenging, remember I'm here to support you.

So our learning outcome today is to be able to use the formula for the cosine rule and the sine rule.

On this slide you can see keywords of things that you have met before but we will be making use of during the lesson.

So let's take a look at them now.

So the sine rule is a formula used for calculating either an unknown side length or the size of an unknown angle.

And you can see it's written there.

The cosine rule is a formula used for calculating either an unknown side length or the size of an unknown angle.

And again, you can see it written there.

For both of these rules, a is the length of the side opposite angle A, b is this length of the side opposite angle B, and c is the side opposite angle C.

So this lesson is about using the sine and cosine rules that we just looked at on the keyword slide.

On our first learning cycle, we're gonna look at using trigonometry with 3D shapes.

And then when we get to the second learning cycle, we'll be looking at bearings.

But for the moment, let's move on with looking at using trigonometry with 3D shapes.

So on the screen, you can see a cylinder with a triangle marked inside of it.

And Izzy has pointed out that now we have the tools to handle non right-angled triangles, we can use that knowledge in 3D shapes too.

Izzy asks herself the question, how much information do I need to find the volume of this cylinder? Pause the video and think about how you calculate the volume of any cylinder.

When you're ready to move on, press play.

So to calculate the volume of a cylinder, we need the radius and we need the vertical height.

The formula for the volume of a cylinder is pi r squared h, where h is the vertical height and r is the radius of the circular face.

So with this triangle inside the cylinder, do we have sufficient information to work those two things out, to work out the radius and to work out the vertical heights? Well, currently we only have an angle between two of the edges and that is not sufficient information to use any of our trigonometric rules.

What about this? Is this enough? So now we have the length of the radius, it's three centimetres and that is one of the edges of this triangle as well.

So do we have enough information? Pause the video and think about that before we continue with this lesson.

So this isn't enough information to be able to calculate the volume of this cylinder.

Izzy points out that normally we need at least three measures.

So if we've got this additional edge, so this edge of 14 centimetres now as well, we now have three measures, two sides and the angle.

But is this enough information to get our volume of our cylinder? Izzy says can I solve it now? What tools should I use? So pause the video and think about if have we now got enough information on that triangle to be able to calculate the vertical height of the cylinder, which is the one measure we need in order to get the volume because we already have the radius.

Press play when you're ready to see what Izzy thinks.

So Izzy's going through her trigonometric rules.

So this is the cosine rule.

So do we have the right information to be able to use the cosine rule? Well the cosine rule can be used if we have two adjacent edges and the angle between them.

Well, we don't have that.

Or if we have all three edges of the triangle.

And we don't have that either.

So the cosine rule is not available as a tool at this current stage.

So let's think about the sine rule.

So for the sine rule we need to have a pair of corresponding edge and the angle.

Do we have that? Or the sine rule would help because we can find all the angles and sides for the triangle because we have that pair of corresponding edge and angle.

So we could use that pair and the three centimetre to work out the top angle of that triangle.

We could then calculate the third angle because we can do angles in a triangle sum to 180 degrees.

And we could then use the sine rule once again to get the third edge.

Or at that point, we could use the cosine rule.

But Izzy points out that actually all we need to get the volume, let's go back to the reason we are talking about this is to get the volume of the cylinder is the height of the cylinder.

Because we have the radius, that three centimetre edge of the triangle is also the radius of the cylinder.

So Izzy has considered this triangle, maybe I just need to use this triangle here.

A radius is a fixed length the whole way round, so that would also be three centimetres on the other side of the centre to the circumference.

The height of that triangle is what we're looking for and we know the edge of 14.

So what could we do with this triangle? Pause the video and think about if this has enough information to allow us to get the volume of the cylinder.

Press play when you're ready to talk about that.

So this one, we do have enough information.

Using this triangle, we can calculate the height of the cylinder using Pythagoras theorem because this would be a right angle triangle.

And then we would have our radius and our vertical height which would allow us to get the volume.

Okay, so let's look at another scenario.

We have a cylinder once again, we're trying to calculate its volume, and we have this triangle and these three measures.

So if we have these three measures, what can we do in order to work out the volume of the cylinder? Pause the video and think about that before we discuss that together.

Izzy says, I think this time I might not have a choice and have to use non right-angled trigonometry.

Did you think about that? So if we use cosine rule, we can calculate the radius, where the radius is the edge length opposite the 15 degree angle.

We have two sides that are adjacent to each other and the angle between them, which means we can make use of the cosine rule.

So Izzy has done just that and she's calculated that the radius of this cylinder is 3.

06 to two decimal places.

She then states, now I can use Pythagoras theorem to find the height then find the volume of the cylinder.

So we can use the Pythagoras theorem with the 10 as our hypotenuse.

We now have that shorter edge which is the radius of the cylinder and we can calculate the height.

So pause the video and work out the height of this cylinder.

Press play when you're ready to move on.

So the height of the cylinder is C to two decimal places.

So you'll have need to work out the radius using the cosine rule, and then use Pythagoras theorem with the 12, and your radius length to work out the vertical height.

We're now up to the only question in task A and in this question you need to find the radius and height of these cylinders to one decimal place.

So on part A, you've got two sides adjacent to each other and the angle between them.

On part B, you also have two sides adjacent to each other and the angle between them.

And in part C, you've got two angles and one edge of the triangle.

You can extend this by then working out the volume if you so wished because once you've got the radius and the height, then you can work out the volume of the cylinder.

So pause the video whilst you're getting at least the radius and height for each cylinder.

And when you press play, we'll go through our answers.

So the answers are on the screen.

The radius, 9.

6, the height, 17.

6, both to one decimal plate.

On B, the radius is 2.

5 and the height is 7.

6 again to one decimal place.

And then for C, the radius is 4.

0 and the height is 17.

2 to one decimal place.

A and B were very similar to the examples we saw in the explanation slides, where you'd need to use the cosine rule to work out the radius and then Pythagoras theorem to work out the height.

Part C was slightly different because we didn't have the same given information.

What we did have was a pair of corresponding edge and angle so we could make use of the sine rule.

We could work out the third angle in the triangle just by doing angles adding to 180 degrees within a triangle and therefore it would be 10 degrees, the missing angle.

So with the sine rule we can work out all of the additional edge lengths and then we can use Pythagoras theorem to get the vertical height.

So we're now at the second learning cycle where we're still working with the sine and cosine rules, but this time we're looking at trigonometry with bearings.

So there are other areas of maths where forming triangles and applying the rules of trigonometry can help us solve problems. So here we've got a diagram with an aeroplane and a helicopter, a north line and an angle mark.

So this is a bearing.

That angle mark is actually a bearing because it's been measured from the north in a clockwise direction.

So we're gonna go through an example of this idea of using trigonometry with bearings, and a similar idea with aeroplanes and flying.

So a plane leaves an airport and travels 12 kilometres north and then seven kilometres northeast.

Calculate A, it's distance from the airport, and B, the bearing it now has to fly to get back to the airport.

So this diagram is been drawn to indicate what has been said in the example.

So it leaves the airport, so we can see that airport is the dot, travels 12 kilometres north, and then seven kilometres northeast.

Well the direction of northeast is 45 degrees from north or the bearing of 45 degrees.

And so that's what we can see with that acute angle.

And then we've got the final point of the plane at that moment in time.

So to calculate its distance, it's the length that's marked as a, so that's the direct distance from the airport to the plane.

And we can use the cosine rule for this.

So why is it the cosine rule? Well we've got two adjacent edges of that triangle, the 12 kilometres north and the seven kilometres northeast.

The angle between them can be calculated because we know that angle from the north line in the direction of the aeroplane travelled northeast was 45 degrees.

So we can calculate that the obtuse angle would be 135 degrees.

And then we have the angle between the two sides.

So to calculate that length, the distance from the airport, we can make use of cosine rule.

Substitute the values in and solve it.

A is 17.

66 to two decimal places and our unit is kilometres.

Part B was the bearing it now has to fly to get back to the airport.

So the arc is on this diagram to indicate the angle we are trying to calculate.

So we need to think about what we have, what we need, and how to get that.

So what can we work out? Well we can work out that the angle from the north line, the second north line anticlockwise is 135 degrees 'cause that's a co-interior angle.

And we could subtract that from 360.

Then we would still have too much.

That's too much of a turn.

The bit we do not need is the angle inside of the triangle.

So we need to be able to calculate that angle.

So we know that angle is 135 degrees and you could have done it using alternate angles equal in parallel lines or co-interior angles sum to 180 degrees.

But regardless of how you got that angle there is 135 degrees, as I say, we need to work out the angle within the triangle to be able to subtract that from 360 degrees as well.

So you can see we're using the sine rule because that angle at A within the triangle has an opposite edge of 12.

What we also need is a pair of corresponding edge and angle, which we have from our calculated distance from part A and the angle of 135.

So using the sine rule, rearranging it to get the angle of A is 28.

72, but that's not yet the bearing.

So being careful, especially when there's multi stages to a problem that we don't find ourselves thinking we must have finished.

We haven't, go back to the question, read it again.

What are we looking for? We're looking for the bearing it now has to fly to get back to the airport.

So 360 degrees is the angles around a full turn minus the 135 minus the calculated 28.

72 is 196 degrees.

And remember that bearings are given as three figures.

So here is a check for you.

Which diagram represents a plane leaving an airport and travelling 22 kilometres north then 40 kilometres northeast.

So pause the video whilst you think about that.

And when you're ready to check, press play.

So B is the correct diagram.

A has got the 22 kilometres and the 40 kilometres in the right way round 'cause it was north and then northeast.

But the angle for northeast is 45 degrees not 40.

Okay, so we have this diagram.

Which calculation we'll find b squared? So pause the video, look at the three equations, and decide which one is correct to work out b squared for this diagram.

Press play when you're ready to check.

So the correct one is C.

So remember this is using a cosine rule, which is a squared equals b squared plus c squared minus 2 times b times c times cosine of a.

But in this case if we're looking for b, then we might relabel it as b squared equals a squared plus c squared minus 2ac cosine of b.

And so 2 times 20 times 15 is 600 which is why A is incorrect.

And the reason B is incorrect is 'cause it's got plus 600 cosine 135 when the formula is minus 2bc equals a or 2ac equals b.

So we're up to the final task of the lesson.

So on question one there are three parts.

So it says help each plane and that's P1, P2, and P3 get back to the airport they start from.

Draw a suitable diagram and calculate the missing information to one decimal place where appropriate.

So on part a, plane one travels north 12 kilometres, then northeast 20 kilometres.

Plane one is at a bearing of what to the airport and how far away in kilometres? So do draw yourself a diagram that really will help you to find the triangle.

Remember we now have the skills, we don't need to have a right angle triangle, we can deal with non right-angle triangles.

On part B, plane two travels northeast 12 kilometres then north 14 kilometres.

Again, what is the bearing to the airport and how far away is the plane? And on part C, plane three travels south 9 kilometres then at a bearing of 36 degrees for 17 kilometres.

So that word is slightly different from the examples and the previous two parts.

So really think about that diagram.

Plane three is at a bearing of what to the airport and how far away? So pause the video whilst you work on question one.

When you press play, we'll move to question two.

Question two, help each boat this time, so boat one and boat two get their bearings.

Draw a suitable diagram and calculate the missing information.

Boat one travels north 12 kilometres, then turns clockwise and travels 14 kilometres to point x, then turns again and travels 20 kilometres before arriving back at the dock.

What was the bearing of boat one at point x from the dock? So draw that diagram carefully.

Read over it a couple of times, check your diagram does accurately show the information given and work out the bearing.

And for part B, boats two travels north 23 kilometres, then turns anticlockwise and travels 16 kilometres to point y, then turns again and travels 18 kilometres before arriving back at the dock.

What was the bearing of boat two at point y to the dock? So pause the video work on question two and then when you're ready for your answers for task B, press play.

So here are the answers.

So you obviously had some gaps to fill in.

So on part A, plane one was at a bearing of 208 degrees to the airport and was 29.

7 kilometres away.

For part B, plane two was at a bearing of 201 degrees to the airport and 24.

0 kilometres away.

Part C, plane three is at a bearing of 245 degrees to the airport and 11.

1 kilometres away.

Onto question two, what was the bearing of boat one at point x from the dock? 44 degrees, but remember you needed three figures so 044.

For part B, what was the bearing of boat two at point Y to the dock? 136 degrees.

So to summarise today's lesson on using the sine and cosine rules.

The cosine rule is useful when you know three sides and an angle are involved.

So this is for non right-angle trigonometry, so non right-angled triangles.

If you have three sides and you're trying to calculate an angle, use the cosine rule.

If you have two adjacent sides and the angle between them, you can use the cosine rule.

And then the sine rule is useful when you know two pairs of sides and angles are involved.

So you can only use the sine rule if you do have a corresponding pair of side and angle.

And if you then have another angle and you need to find the corresponding edge to that, or if you have another side and you're trying to work out the corresponding angle to that.

So well done today and I look forward to working with you again in the future.