Lesson video

Hello, my name is Mr. Clasper.

And today we're going to be using the cosine rule to find missing lengths on a non-right angle triangle.

Before we begin this lesson, if you haven't done so already, I would recommend looking at the lesson on using the sine rule to find either a missing length or an angle as this has some ideas linked to labelling triangles, which are very important when we use advanced trigonometry.

In today's lesson, we're going to use the cosine rule.

The cosine rule states that a squared is equal to b squared plus c squared minus 2bcCosA.

When we use the cosine rules, labelling is especially important.

We need to make sure that we label the angle A carefully as it's going to be involved in our working to find the side of a.

Now, to find where the angle A must go, we look at our triangle and we're looking at the side we need.

So we're missing the side y and it's the angle opposite to this, which must be angle A.

So we're going to label the angle of 40 degrees as angle B.

Angles B and C can go on the other two vertices and if we label our sides with lowercase a, b and c, we can then begin to use the cosine rule.

Alternatively, as long as this is angle A, we could label B and C the other way around.

As long as we label our sides appropriately, this won't have an impact on our final answer.

If I move the 40-degree angle, that means my new 40-degree angle must now be angle A.

And again, B and C can go on either of the other two vertices, providing that the sides are labelled correctly.

So we have lowercase a opposite angle A, b and c.

Take a look at this triangle.

Can you label this triangle correctly so that the cosine rule can be used to find the side y? Pause the video to complete your task and click Resume once you're finished.

Did you get it correct? So just remember, the angle A must be opposite the side you are looking for.

You could have also had this labelling instead.

Again, the angle A is in the right place and it doesn't matter which way round B and C go.

Calculate the length of the side y.

Round your answer to three significant figures.

Now, we're going to use the cosine rule and we need to remember that angle A must be opposite the side we are looking for.

Therefore, angle A must be our 42-degree angle.

Angles B and C could be labelled here.

And the sides a, b and c lowercase could be here, here and here.

Now, when we look at the rule, all we need to do now is to substitute any information we know.

So the only piece of information we don't know is the value of a.

However, we know that a squared must be equal to seven squared plus nine squared minus two multiplied by seven multiplied by nine multiplied by Cos42.

And if I calculate the value of all of this, it tells me that a squared must be equal to 36.

3637.

Now, if we look carefully, we need to find the value of a.

We currently have the value of a squared.

So what we need to do is to square root our answer.

If we square root, we will find the value of a.

So using the answer button on your calculator, the ANS button, you could press the square root button, followed by the ANS button and this will give you the square root of 36.

3637, which is 6.

0302.

If I refer back to the question, it says I need to round my answer to three significant figures.

So to three significant figures, the side y would be approximately 6.

03 centimetres.

Let's try this example.

Calculate the length of the side BC.

Round your answer to one decimal place.

We're going to use the cosine rule again and remember, labelling is important.

However, our angle A is already opposite the side BC.

So we can leave our angles as they are and label our sides with lowercase a, b and c.

Once we've done this, we can substitute into our formula.

And calculate in the right-hand side, this means that a squared must be equal to 143.

6139.

Now, as this is the value of a squared, we're going to need to square root to find the value of a.

That means that a must be 11.

9839.

If I refer back to the question, it says round your answer to one decimal place.

Therefore, the length of BC must be 12.

0 centimetres.

Here is a question for you to try.

Pause the video to complete your task and click Resume once you're finished.

And here is your solution.

So remember, make sure you label your triangle carefully, particularly paying attention to angle A, which must be opposite the side you are trying to work out.

Once you've done that, the last step, just remember to square root once you've substituted everything into the formula so that you get your correct value, which should be, in this case, 26.

7 metres.

Here's another question for you to try.

Pause the video to complete your task and click Resume once you're finished.

And here is your solution.

So once again, just take care with your labelling.

Make sure that angle A is labelled opposite the side which we want.

So it should be opposite the side AB.

And once we've done that, take care with your substitution, make sure you square root at the end and we should get a value of 20.

73 metres.

Here's a question for you to try.

Pause the video to complete your task and click resume once you're finished.

And here is the solution.

So if we look carefully at the working out, Mo has actually squared 15 and 11 twice in his calculation.

He should have calculated 15 squared plus 11 squared minus two multiplied by 15, multiplied by 11 multiplied by Cos71.

If he did this, and square rooted his answer, that would then give him 15.

4 centimetres.

Let's have a look at this example.

Calculate the perimeter.

Round your answer to three significant figures.

So I need the perimeter of this triangle.

However, I'm missing one of the three sides.

I can use the cosine rule to find my missing side.

So if I label my triangle with A, B and C again, my angle A is opposite the side I'm missing and if I label my sides, I can substitute the information into my formula.

This means that BC squared must be 24.

4259.

However, I need BC, so I'm going to need to square root both sides.

So if I take the square root of 24.

4259, this leaves me with 4.

9422.

And to three significant figure, this would be 4.

94.

So now we know that the length of BC is 4.

94 centimetres.

Now, if we use this for the remainder of the problem, to find the perimeter, I need to add the three sides of my triangle together.

So 4.

94 plus nine plus six is equal to 19.

94.

Therefore, my perimeter must be 19.

9 centimetres, correct to three significant figures.

Here's your last question.

Pause the video to complete your task and click Resume once you're finished.

And here is your final solution.

So if we take a look at our diagram, we need to find the length YZ as this is missing, and we're going to need it to calculate our perimeter.

So to do that, we need to label our triangle, making sure that the 45-degree angle is labelled as A and we can substitute into our formula.

We should find that this length is 36.

911 metres long.

Now that we know this, we need to find the perimeter of the triangle.

So we found a missing side, but we need to find the perimeter.

So to do this, we need to add the other two sides, so we need to add 40 and 52 to our 36.

9 and we should find that the perimeter is 128.

9 metres long.

And that brings us to the end of our lesson.

So you've managed to use the cosine rule to find a missing length in a non-right angle triangle.

That's brilliant.

Why not try the exit quiz to show off your brand new skills? I'll hopefully see you soon.