Lesson video

Hi, I am Mrs.Danny and I'm really pleased that you've been able to join me for this lesson.

Because this is one of my favourite math topics.

We're going to be looking at the trigonometric ratios for values of zero, 30, 45, 60 and 90 degrees.

Let's go.

We use two different triangles to help us find the exact values for sine, cosine and tangent.

When we have an angle of 30 degrees, 45 degrees or 60 degrees.

This equilateral triangle with side lengths of two centimetres, can be used to find the values for the trigonometric ratios at 30 degrees and 60 degrees.

We split the equilateral triangle into two identical or congruent right angled triangles.

Let's look at one of these triangles in more detail.

And work out its missing side length and missing angles.

We know that it's longest side or it's hypotenuse is two centimetres.

The base must be one centimetre.

As it is half of the base of the equilateral triangle.

We need to find the perpendicular height x.

We do this using Pythagoras' theorem.

Square the smallest side lines and add them together to get the square of the hypotenuse.

We can then rearrange this to find x.

So x must be root three.

We now need the two missing angles.

Well, all angles in an equilateral triangle are equal.

So that means that they are 60 degrees each.

So the bottom right angle in our smaller triangle must be 60 degrees.

The top angle is half of the top angle in the equilateral triangle.

So this has to be 30 degrees.

We needed a triangle with 30 degrees and 60 degree angles.

So now we can use this, to find the exact trigonometric values for sine, cos and tan.

For sin 30, we want the opposite side divided by the hypotenuse.

The opposite side is one centimetre and the hypotenuse is two centimetres.

So the exact value for sin 30 is a half.

You can use this triangle to find all the other exact values for sin 60, cos 30, cos 60, et cetera.

Now it's your turn to use the triangle to find some other exact values.

Pause the video to complete the task and restart when you are finished.

Here are the answers.

Take care to label the opposite side correctly.

Depending on whether you're using the 30 degree angle or the 60 degree angle.

Next, we want to work out the exact values for sin 45, cos 45 and tan 45.

Here's another triangle that can help us to do this.

It's a right angle isosceles triangle with a base and perpendicular height of one centimetre.

We know the angles are 90 degrees, 45 degrees and 45 degrees.

So we only need to find the length of the hypotenuse x.

We use Pythagoras' theorem again to help us to find x.

And we find that x is equal to root two.

Now we can work out some exact values.

What is tan 45? Well tan is the opposite divided by the adjacent.

So this is one divided by one.

Which is equal to one.

So tan 45 is equal to one.

Here's a question for you to try.

Pause the video to complete the task and restart when you are finished.

Here are the answers.

Notice that sin 45 is equal to cos 45.

And tan 45 is equal to one.

So you might find these values a little bit easier to remember.

Without having to draw out the triangle each time.

Here's a useful table of exact trig values for you to complete.

Can you manage to find the exact values for zero degrees and 90 degrees? Try to draw some right angle triangle diagrams and think about what is happening to each side length and the trig ratios as the angle theta changes.

Pause the video now to complete the table and restart when you are finished.

Here are the answers.

Did you manage to get some of the exact values for zero and 90 degrees.

Here's an example for zero degrees.

As theta gets closer to zero, the opposite side gets shorter and shorter.

And the hypotenuse becomes the same length as the adjacent side.

So for sin zero degrees equal to the opposite over the hypotenuse, we get zero, divided by whatever length the hypotenuse is.

Zero divided by any number is always zero.

So sin zero degrees is equal to zero.

I'll let you investigate the other ratios.

But I'll just draw your attention to tan 90 degrees.

This value is undefined.

Because we would end up with tan 90 equals the opposite divided by the adjacent.

And the value for the adjacent will be zero.

So we will be dividing the opposite by zero.

Division by zero is undefined.

So tan 90 degrees is therefore undefined.

We're now going to use the exact trig values to work out an angle without a calculator.

Label the sides of the triangle first.

We have side lengths on the hypotenuse and adjacent sides.

So we use cos.

Cos theta is equal to the adjacent divided by the hypotenuse.

So cos theta equals seven divided by 14.

Seven fourteenth simplifies to a half.

So we can either use our table from earlier or the triangles we drew earlier to find out when cos theta is equal to a half.

This happens when theta is equal to 60.

So theta is 60 degrees.

Here are some final questions for you to try.

Pause the video to complete them and restart when you are finished.

Here are the answers.

Did you manage to remember any of the exact trig values? It is useful if you can learn these by heart.

But if like me, your memory isn't that great.

Just draw the equilateral triangle or the right angled isosceles triangle and find the missing sides and angles.

That's all for this lesson.

Remember to take the exit quiz before you leave.

Thank you for watching.