Lesson video

Hello, my name is Mr. Clasper and today we are going to be using the cosine rule to find a missing angle inside a non right angle triangle.

In this lesson we're going to learn how to use the cosine rule to find a missing angle.

In the previous lesson, we used the cosine rule to find a missing side and we were given the rule that a squared is equal to b squared plus c squared minus 2bc cos A.

Now to use this to find missing angles, it's often useful to rearrange the formula.

So let's have a look at this.

This is our original rule, and first of all I'm going to add 2bc cos A to both sides of this formula.

That must mean that a squared plus 2bc cos A must be equal to b squared plus c squared.

Now I'm going to subtract a squared from both sides, this means that 2bc cos A must be equal to b squared plus c squared minus a squared.

And if I divide both sides by 2bc, this means that the cos of angle A must be equal to b squared plus c squared minus a squared all over 2bc.

And this is the rule that we're going to use today.

Let's have a look at this question.

Calculate the size of the angle y, round your answer to three significant figures.

So I need to label my triangle first.

Now again, when I label I need to make sure that the angle I'm looking for is angle A.

And angles B and C can go on either of the remaining two vertices.

I need to make sure that I label my sides appropriately.

So a, b, and c.

And from here I can input my information into the rearranged cosine rule.

So if I calculate seven squared plus nine squared minus eight squared, and divide this by two multiplied by seven multiplied by nine, this will give me the value of cos y.

Now the cosine of y is equal to 0.

5238, but I want the value of y.

So I'm going to need to take the inverse cosine of 0.

5238.

This will give me a value of 58.

4118.

If I refer back to the question it says round your answer to three significant figures, therefore my final answer would be 58.

4 degrees for the size of angle y.

Let's try this example.

Calculate the size of angle YXZ.

Now when we use this rule again, it is very important where we label angle A.

So angle A must be where YXZ is.

So that means this angle is angle A, and this can be angle B and C.

Now I've done this, I just need to label my sides appropriately.

So lowercase a, b and c.

And now I can substitute any information I have into the given formula.

If I calculate the right hand side, this means that the cosine of angle y, x, z must be equal to negative 0.

2482.

However I don't want the cosine of this angle, I want the value of the angle itself.

Therefore I need to take the inverse cosine of my value of negative 0.

2482.

If I do this, I then get a value of 104.

37408.

This means that y, x, z is approximately 104.

4 degrees.

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 your solution.

So if you look at our example, we've labelled the angle we are seeking, x, y, z as angle A, and labelled appropriately from this point.

From here, it's a case of substituting your values into your rearranged formula.

Now remember, this rearranged formula will give you the value of cos x, y, z.

So to get your final answer you need to take the inverse of the value you receive from the right hand side of the equation.

Therefore x, y, z must be 30.

2 degrees.

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 your solution.

So once again, take care with your labelling.

So angle y was labelled as angle A, as this is the angle we are seeking.

And we've substituted it.

Again remember to take the inverse cos of the answer you get from your calculator.

And this will get you y must be equal to 129 degrees.

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 if we look at the two cards, they both look very similar.

However, if we look at the top cards, we can see that this has got a calculation of 26 squared plus 17 squared plus 32 squared.

But our formula is that cos A is equal to b squared plus c squared minus a squared all over 2bc, therefore the second card is the correct answer.

Here is your last question.

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

And here is your final solution.

So we were asked to calculate the smallest angle in the triangle.

Now once that looks as though why YXZ is the smallest angle, it wouldn't hurt to check the other two as well.

So you may need to use the cosine rule three times with three sets of labelling to establish that YXZ is indeed the smallest angle.

Once you've done that, we just need to substitute carefully and we should find out that the smallest angle, YXZ is equal to 37.

7 degrees.

And that brings us to the end of our lesson.

So you can now use the cosine rule to find a missing angle.

Go you! Well now have a go at the exit quiz to show off your brand new skills.

I'll hopefully see you soon.