Lesson video

Hi, I'm Mr. Bond, and in this lesson, we're going to learn how to use a combination of the sine and cosine rules, and also the area formula, 1/2 ab sine c, in order to solve some problems. In this lesson, we're going to be looking at problems involving the sine rule, the cosine rule, and the area formula for a triangle, area is equal to 1/2 ab sine C.

Let's start with a quick recap of when we can use the sine rule and cosine rules.

So we can use the sine rule if we know two angles and a side or two sides and a non-included angle, and we can use the cosine rule if we know three sides or two sides and the included angle.

Here's our first example.

We are asked to work out the area of the triangle.

So the first thing that I do when looking at a problem like this is think, "Am I given values for the base and perpendicular height? Because if I am, that makes things very simple." In this case, I'm not.

I'm given values for the lengths of two sides and an angle.

And the two sides aren't perpendicular to each other.

So I'm going to have to use the area formula, area is equal to 1/2 ab sine C.

So let's label our triangle up with lengths a, b, and c.

So I know lengths a and b, but I don't know the angle C.

Can I work it out from the information that I'm given? Well, yes.

I'm given two sides and an angle.

This means that I can use the sine rule to find another angle, but is it angle C? No.

We're going to find angle B and then use the fact that angles in a triangle sum to 180 degrees.

So to find angle B using the sine rule, we can say sine B over 14 is equal to sine 37 over nine.

Multiplying both sides of this equation by 14 gives us sine B is equal to 14 sine 37 over nine.

And then using the inverse sine function, B is equal to the inverse sine of 14 sine 37 over nine.

Using our calculator to calculate this will give that B is equal to 69.

415829 degrees.

So we can then use this to find angle C, which will be equal to 73.

584171 degrees.

Now that we know this angle, we can use the area formula.

The area is equal to 1/2 ab sine C.

Substituting our values for lengths a and b and the angle C gives this, the area is equal to 1/2 multiplied by 14 multiplied by nine sine 73.

584171.

You could use the memory function on your calculator to store this angle.

When we perform this calculation, it will give that the area is equal to 60.

4 millimetres squared.

Here's a question for you to try.

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

Here are the answers.

Part a was quite straightforward.

All we had to do was use the area formula, the area is equal to 1/2 ab sine C, in order to find the area.

There were many ways that potentially we could have done part b, but the most straightforward way would have been to use the cosine rule, because we know two sides and the included angle and we're looking for that third side.

Here's another question for you to try.

Again, pause the video to complete the task and resume the video when you're finished.

Here's the solution to question number two.

The way that I approached this question was to find the third angle in the triangle by knowing that angles in a triangle sum to 180 degrees, and then I found the length of the side XY using the sine rule.

Once I knew this, I found the area using the formula the area is equal to 1/2 ab sine C.

You might have used a slightly different way.

If you have, all that we should really consider, just like I consider every time I solve a problem, was is this accurate, and was it the most efficient way that I could have solved the problem? Here's another example.

We have to work out the area of the quadrilateral ABCD.

Rather handily, this quadrilateral has been split into two triangles.

So if we can find the area of both of these triangles, we can simply add them together to find the area of the quadrilateral.

Which of the two triangles will it be easiest to find the area of? Hopefully, you've spotted that this is triangle ABC.

It's easier to find this area currently, because we know two lengths and the included angle.

So we can use the formula that the area of triangle ABC is equal to 1/2 ac multiplied sine B.

This looks a little bit different to how we've used the area formula before.

This is because the included angle is angle B in this case and the two edges on either side are A and C.

So substituting into this formula gives that the area is equal to 1/2 multiplied by 12 multiplied by 11 sine 83.

And if we put this into our calculator, it will give 65.

50804.

So this is the area of triangle ABC.

Now, let's turn our attention to the other triangle.

Can we find the area of this triangle in the same way? Well, we could if we knew the length AC.

So to find length AC, we're going to use the cosine rule on triangle ABC.

So the cosine rule is given by b squared is equal to a squared plus c squared subtract two ac cos B.

Substituting into the formula for values a, c, and the angle b gives this.

And when we put this into our calculators, it tells us that b squared is equal to 232.

826 et cetera.

So when we take the square root of this, it gives that b is equal to 15.

258 et cetera.

Now that we know the length AC, we can find the area of triangle ACD using the area formula.

The area of ACD is equal to 1/2 ad multiplied by sine C.

Substituting into this formula gives 1/2 multiplied by 14 multiplied by 15.

258 et cetera multiplied by sine 58.

And this gives 90.

580501.

Notice how I haven't rounded either of the areas yet.

That's so that my answer is as accurate as possible and I'll only do any rounding at the final stage.

Now that I know the areas of both triangles, I can add these together to find the area of the quadrilateral, which is equal to 156 metres squared to three significant figures.

Here's another question for you to try.

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

Here's the answer to question number three.

To answer part a and find the length BC, I used the fact that I knew the area of triangle ABC and then substituted in the values for the length AB and 35 degrees and then solved to find the missing length BC.

Then, we had to calculate angle BCD.

We know three sides by this point in the question, so we can use the cosine rule.

And here's the final question for this lesson.

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

Here's the answer to question number four.

To start with in this question, I found the length of the side WY.

I did this by using the sine rule, because I knew two angles and a side for the triangle WYZ.

Once I knew the length of WY, I know all three sides of triangle XWY, so I can use the cosine rule to find any of the three angles.

That's all for this lesson.

Thanks for watching.