Lesson video

Hello, my name is Mr Clasper.

And today we're going to be looking at the sine rule, and using this to find a missing angle inside a non-right angled triangle.

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

If you haven't done so already, I would recommend watching the lesson on using the sine rule to find a missing length.

In that lesson, we looked at the sine rule, which has given in this box.

What we're going to use today, is a re-arranged form of this rule, which looks like this.

Now, if you look closely you can see, that the denominators have become the numerators, and the numerators have become the denominators.

If we think about this, it does make sense.

So if I said that, the side a divided by the sine of angle A, had a value of three over five, that would mean that the side b, divided by the sine of B would also be equal to three over five.

And the side of c, divided by the sine of angle C, would also have a value of three over five.

And it makes sense that, this must mean that the sine of angle A, divided by a must be, five over three.

And therefore, the sine of angle B, divided by the side of B would also have a value of five over three, and so on.

I'm going to use this rearrangement to help me, find a missing angle.

Let's look at this problem.

To calculate the size of the angle y, my first step is to label my triangle.

So I'm going to use upper case letters for my angles, A, B and C and lower case letters for my sides a, b, and c I'm then going to use my formula and substitute information that I know.

So I know the value of the angle A, and the side of a, and I also know the value of the side c.

So I can substitute this information in.

Now to isolate, y I can multiply by 12 on both sides of my equation.

This leaves me with 12 multiplied by sine 42 over nine, and this is equal to the sine of y.

This gives me a value of, 0.

89217.

Now, does it seem reasonable that the angle y is less than one degree? I don't think so.

So let's take a look at this.

This is actually equal to the sine of y, but we need to find the value of y.

So in other words, the sine of something which would give us 0.

89217.

To do this, we need to do the inverse sine operation, of 0.

89217.

To access this function, press Shift on your calculator, followed by the sine function, and then input the number.

Alternatively, if you use the button, which has Ans written on it, this will solve a lot of problems for you.

This button will remember your previous answer.

So if you've worked through, this working, your calculator will currently remember the number 0.

89217.

So if you press Shift + sin and answer, it will give you a value of 63.

14773.

So that means that our angle y is approximately, 63.

1 degrees.

Let's try this example.

We're going to find the size of angle z.

So I need to label my angles with upper case, A, B and C, and I need to label my sides with lowercase, a, b, and c.

I can set up my equation and substitute my values, which I know.

From this point, I need to multiply both sides of my equation by 10.

7.

As this will leave me with the sine of z.

And this is equal to, 0.

7083.

Now remember, this is the value of the sine of angle z.

However, we want the value of the angle z.

So we need to take the inverse sine of 0.

7083.

Now remember, if we use the Ans button or the answer button, this will save us a job.

And it gives us a value of 45.

1001, which is approximately, 45.

1 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's our solution.

So again, remember your first step is to make sure, you label your triangle carefully.

Once you've done this, you can set up your equation.

And if you look at the second step of our working out, we get a value of, 0.

482.

Now remember, this is the value of the sine of a, not the value of a.

So you need to take the inverse sine of this value to get your final answer, which is 28.

8 degrees.

Let's try this example, calculate the size of angle XYZ.

Round your answer to three significant figures.

Let's label our triangle first, with upper case letters for angles, A, B, and C, and we're going to use lower case letters to annotate our sides, a, b, and c.

I can set up my equation, and I can input any values I know.

Now to isolate, X, Y, Z, I need to multiply both sides of my equation by nine, which would give me, nine multiplied by sine 42, over seven, which is equal to the sine of angle XYZ.

And this gives us a value of, 0.

8603.

Now remember, we don't want the sine of angle XYZ, we want the value of XYZ.

So we need to take the inverse sine of this value.

This will give us, 59.

3514.

Referring back to the question, it says, round your answer to three significant figures.

So this would give us a final answer of 59.

4 degrees for the size of angle XYZ.

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 make sure you label your triangle carefully.

And again, looking at the second step of working, remember that value is 0.

711.

This is the value of the sine of angle XYZ.

So you need to take the inverse sine of this value to get your final answer, which should be, 45.

3 degrees, rounded to three significant figures.

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 the solution.

So Baz made a horrible error.

He's calculated 13 multiplied by 18, over sine 51.

What he should've done, was to calculate 13 multiplied by sine 51, over 18.

Again, be careful when we set up our equations, and this will avoid this mistake.

Here's your last question.

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

And here is your solution.

So your first step, would be to find your interior angle using the exterior angle given.

So if the exterior angle is 117 degrees, that means our interior angle at C, would 63 degrees.

From here, make sure that you've labelled, your triangle correctly.

And again, we can follow the same procedure.

So we can set up our equation.

And then we can solve that, and we should end up with a value of 52.

4 degrees, to three significant figures.

And that brings us to the end of our lesson.

So today you've learned how to use the sine rule to find a missing angle in a triangle.

I think you should try the exit quiz, to show off your brand new skills.

Take care.