Lesson video

Hello! My name is Mr. Clasper and today we are going to be using the sine rule to find a missing length in a triangle.

In this lesson we're going to learn how to use the sine rule to find a missing length in a non-right-angled triangle.

Before we do this, we need to learn how to label a triangle appropriately so that we can use this formula.

The way we do this is relatively straightforward.

We need to label each of the vertices with A, B and C.

It doesn't matter which vertices is which as long as you have one of each.

What is important is how we label our sides.

So which side we label lowercase a is determined by where the angle A is located.

So the side A is directly opposite the angle A.

And the side B, which we notate with a lowercase b is directly opposite the angle labelled B.

And finally the side C is labelled with a lowercase c and it is directly opposite the angle we've labelled C.

And this is how we label a triangle to use the sine rule.

Have a look at these four triangles, which two triangles have been labelled incorrectly? Pause the video to complete this task and click resume once you're finished.

The red triangles have been labelled incorrectly.

If we look at the top left triangle, this triangle has been labelled with lowercase letters for angles and uppercase letters for sides.

But it needs to be the other way round, so we need to remember that uppercase letters are used for angles and lowercase letters are used for sides.

And if we look at the bottom left triangle, if we look carefully, we do have three angles labelled A, B and C.

However, if we look at the side B, this is not opposite the angle B.

It is opposite the angle C.

Therefore this triangle has been labelled incorrectly.

If we look at the other two examples, we can see there have been labelled correctly.

As the angles are uppercase, and the sides are lowercase.

And each side corresponds to its opposite angle.

Let's take a look at the sine rule.

Thinking about triangles, this would mean that the side labelled A, divided by the sine of angle A is equal to the side label B divided by the sine of angle B.

And this is also equal to the side labelled C divided by the sine of angle C.

So, if we took the length of A and divided it by the sine of angle A, this would give me the exact same answer, as if I divided the side of B by the sine of angle B.

Let's take a look at this example.

Calculate the length of the side y.

Before I do this, I need to label my triangle.

So I'm going to label each vertices with A, B and C.

And I'm going to use my lowercase letters to label my sides appropriately a, b and c.

And now I can move forward.

I'm going to use my sine rule.

I'm going to use the fact that I know that the side divided by the sine of angle A is equal to the side B divided by the sine of angle B.

And now what I'm going to do is substitute any information which I know.

So I know that A is nine.

I know that the angle A is 42.

However, I don't know the length of side B, and I don't know the size of angle B.

This presents a problem as I cannot solve this equation.

It has more than one unknown quantity.

Therefore I cannot solve it.

Let's try this a different way.

I'm going to use the fact that the side A, divided by the sine of angle A is equal to the side C divided by the sine of angle C.

And again I'm going to substitute my information.

So I know that A is nine, and I know that the angle A is 42.

I don't know the length of y, as this is what we're trying to find out.

However, I do know that the angle C is 63.

In my equation, I only have one variable which I don't know the value of yet.

I can try to solve this.

To solve this equation, I'm going to multiply both sides by sin63.

This will give me the value of y.

Now all I need to do is to input nine over sin42 multiplied by sin63 in my calculator.

And when I do this, I should find a value of 11.

9842.

Which would approximately be 12 centimetres.

So the side y is approximately 12 centimetres long.

Let's take a look at this example, we're going to calculate the length of the side z.

I need to label my angles with uppercase A, B and C.

Then I need to label my sides with lowercase a, b, and c.

Now to decide what my equation looks like, I'm going to need to use B and the sine of B as the side B represents the side Z, which I'm looking for.

And I'm also going to use C over the sine of C.

Because I have information about both of these.

Now I can substitute the information I have.

The only piece of information I don't know is the value of z.

And this is a piece of information we're trying to find out.

So as an equation, I can multiply both sides by sine of 47, to isolate the variable z.

And if I calculate 12.

3 divided by sin36, multiplied by sin47, this gives me a value of 15.

30431.

which would approximately be 15.

3 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 the first point to make is that you need to make sure that you label your triangle carefully.

So take note of where the sides must be.

From this point we can set up an equation and we can see in our equation we've multiplied both sides by sine of 100, to get our final answer of 19.

7 centimetres.

Here is 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 like the last example, make sure you label your triangle carefully, and when we're rounding to three significant figures, the first significant figure was seven followed by eight and then three.

So our final answer is 7.

83 metres to three significant figures.

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, the equation that Tamsin should have started with should have been BC over sin50 is equal to 20, over sin95 and rearranging this would mean that we would actually calculate sin50, multiplied by 20, over sin95.

Let's try this example, calculate the length of side y.

We can begin by labelling our triangle.

So label our angles with uppercase A, B and C and label our sides with lowercase a, b and c.

Now in this example, we can't create an equation with one variable.

In every instance, we will always create an equation with more than one variable, therefore, we won't be able to solve it as it is.

If we look closer however, at the triangle, we have two of the three angles inside the triangle.

And we should know that the angles inside a triangle have a sum of 180 degrees.

Therefore this means that our angle A is actually 85 degrees.

Now we have this information, we can move forward with our question.

So, if we take A over sinA and make it equal to C over sinC and substitute our information, we have an equation with one variable which we can now solve.

So if we multiply both sides by sine of 85, we get y is equal to 14 over sin32 multiplied by sin85, and the value of this from a calculator will give us 26.

3185, which is approximately 26.

3 centimetres.

Here is your last question, pause the video to complete your task and click resume once you're finished.

And here is your solution.

So the first thing we needed to do was to find our missing angle.

So you should find that the missing angle XYZ was 60 degrees.

And then using this angle we can find our missing side by using the sine rule.

So our missing side was 35.

863 metres.

From there we can establish that we needed 105.

1 metres, which means that John did not have enough.

And that brings us to the end of our lesson.

So we've been using the sine rule to find a missing length in a non-right-angled triangle.

Why not try the next lesson on finding a missing angle inside a non-right-angled triangle using the sine rule.

You could also try the exit quiz to test out your skills from today.

I'm sure you'll do absolutely fine.