Lesson video

Hello, I'm Mrs. Lashley and I'm gonna be talking you through the lesson today.

I really hope you're willing to try your best and are looking forward to learning something new.

I'll be there to support you as we go through the lesson.

So our learning outcome today is to be able to calculate trigonometric ratios for 0 degrees, 45 degrees and 90 degrees.

On the screen, there are some keywords that I'll be using during the lesson.

I'm gonna run through 'em now to make sure that we are familiar with them.

They're not new to you.

You will have learned them before in your learning, but it's important that we are okay before we start the lesson.

So trigonometric functions are commonly defined as ratios of 2 sides of a right-angled triangle for a given angle.

The sine of an angle, which we use the shorthand notation of sine, which is sin, is the y coordinate of the point P on the triangle formed inside the unit circle and you can see that on the diagram to the right.

The cosine of an angle, which again, we use the first 3 letters of the word, so it looks like cos.

So cosine of theta is the X coordinate of point P on the triangle formed inside the unit circle.

A tangent to a circle is line that intersects the circle exactly once.

And the tangent of an angle is the y coordinate of point Q on the triangle which extends to the unit circle.

And once again, you can see that on the diagram.

So you may wish to pause the video and read the definitions to yourself and cross-reference them to the diagram to ensure you are happy before we move on with the lesson.

So the lesson about calculating trigonometric ratios for 0 degrees, 45 degrees and 90 degrees is split into 2 learning cycles.

The first learning cycle is finding the exact values for those 3 angles, and then later on we're gonna be looking at spotting relationships that are making use of the exact trig values of 0 degrees, 45 degrees and 90 degrees.

So let's make a start at finding the exact trigonometric values for 0, 45 and 90 degrees.

So when we're working with trigonometry, we often only see rounded values and this is because the calculator can only display a certain number of digits.

So if we look at these 2 displays that have come from a calculator, we've got on the left hand side sine of 38 degrees.

We know it's 38 degrees because on the top line of the calculator display you can see the capital D.

That informs us that our unit for the angle is in degrees.

And it gives us the value 0.

6156614753.

So this is sine of 38 degrees to 10 decimal places.

And on the right hand side it's cosine of 12 degrees.

And again there is 10 decimal places, but that's not because the value of cosine 12 or the sine of 38 degrees is equal to a finite 10 decimal places, it's because of the calculator displays limitations.

So there are certain trigonometric ratios that can be expressed exactly.

And Jacob wonders which 1s and how can I tell? Well, 30 degrees and 60 degrees can be expressed exactly and we can do that using an equilateral triangle because of the 60 degree angle that we'll find in every equilateral triangle.

And actually if we use half of it, if we split it into 2 congruent right-angled triangles, then our right-angled triangle has a 60 degree angle and a 30 degree angle.

And then we can find exact trigonometric ratios for 30 degrees and 60 degrees.

But what other ones? Well, as we know today we're looking at trying to find the exact trigonometric values for 0, 45 and 90 degrees.

So how can we do that? So for a check, we are gonna start by considering an isosceles right-angled triangle and you can see it here on the screen.

But what is the length of the hypotenuse? So pause the video and think about how you're gonna calculate the length of that hypotenuse and leaving your answer in its exact form.

Press play when you're ready to check.

So we're gonna make use of Pythagoras' theorem to calculate the hypotenuse.

It's a right-angled triangle, which means that Pythagoras' theorem holds.

So 1 squared plus 1 squared is equal to the hypotenuse squared.

And so if we square root that hypotonus squared we'll get the length of the hypotonus, which is root 2.

So we're gonna leave it in that exact form because we want all of our answers to be exact values.

So we do not want to include a rounded value on the hypotenuse if we're going to make use of this right-angled triangle.

So if it's a right-angled isosceles triangle, then we can also deduce what the 2 remaining interior angles must be.

Pause the video and think about what they are.

And then as when you press play, we'll check that you've got those right and we'll move on.

So it's gonna be 45 degrees each and that's because it's an isosceles triangle.

So isosceles triangles have 2 equal angles.

And if 90 degrees of the 180 degrees has already been used by the right-angle, there's only 90 degrees remaining.

And if they have to be equal, when we split that into 2 equal parts, we'll have 45 degrees each.

So now that we have this right-angled iso leaves triangle where we have 3 exact edge lengths and 3 angles, we can use this to deduce exact values for the trigonometric ratios when the angle is 45 degrees.

So just take a moment to think about what the trigonometric ratios are, which 2 side lengths are important for each one.

Press play when you're ready to move on.

Hopefully, you've just thought about sine and cosine and tangent and what the ratio is.

So what is sine of 45 degrees? Use the diagram to support you to get the exact trigonometric value.

Press play when you're ready to check your answer.

So the answer is c, root 2 over 2.

You may have got 1 over root 2 using the diagram because the opposite of any of the 45s is 1 and the hypotenuse is root 2.

And that is equivalent to the answer c, root 2 over 2.

But just to remember that we don't tend to leave our fractions with an irrational denominator.

We rationalise the denominator.

So sine of 45 degrees is equal to root 2 over 2.

Okay, another check.

What's the exact trigonometric value for cosine 45 degrees? Once again, pause the video, use the diagram and when you're ready to check press play.

Cosine 45 degrees is also 1 over root 2, but we know that that is the same as root 2 over 2.

Why is it the same? Well, because the adjacent of 45 degrees is also 1 because this is an isosceles triangle.

Finally, what is tangent of 45 degrees? Pause the video and then when you're ready to check, press play.

Tangent of 45 degrees is 1 because the tangent ratio is the ratio between the opposite and the adjacent.

And in this particular triangle, both the opposite and the adjacent are equal because it's an isosceles and therefore 1 over 1 is equal to 1.

We're up to the first task of today's lesson and both questions are on the screen here.

In question 1, you need to complete the table with the exact values.

So that means you shouldn't be rounding anything, you shouldn't be using your calculator and find in a decimal.

We should be deducing them using the equilateral triangle for 30 degrees and 60 degrees and the Right-angle isosceles triangle for the 45 degrees.

When you get to question 2, you're gonna justify through reasoning the values in the 0 degree and the 90 degree column in question 1.

So they've been filled in already for you.

But for question 2 you're gonna justify why they are the values.

So pause the video and when you press play we'll go through the answers to Task A.

So here are the answers to question 1 of Task A and you can see the exact values have been filled in.

So for the 45 degree column, that middle column, you should have sine of 45 degrees is root 2 over 2.

cosine of 45 degrees is root 2 over 2 and tangent of 45 degrees is 1.

And we can deduce that using the right-angled isosceles triangle.

For sine of 30 degrees, it's 1/2, for sine of 60 degrees, it's root 3 over 2.

Cosine of 30 degrees is root 3 over 2, cosine of 60 degrees is 1/2.

Tangent of 30 degrees is root 3 over 3 and tangent of 60 degrees is root 3.

For question 2, you are justifying through reasoning the values in the 0 degree column and the 90 degree column.

So this is looking at the 0 degree column to begin with and I've justified it using the unit circle diagram.

So using the unit circle, we can see that when the angle is 0 degrees, sine of theatre is 0, cosine of theatre is 1 and tan of the is also 0.

But where does that come from? Well, when we look at the unit circle, we are looking at the angle of rotation of the radius of the unit circle from the positive X axis.

So if the angle of rotation is 0, the radius is horizontal and lying on the X axis because there is no angle of rotation.

So if we remember from the definitions and the keyword slide, the point P which is at the end of the radius and on the circumference of the unit circle will be the point 1, 0.

And in this particular case, the point Q is equal to the point P and that was if you remember the tangent where the line the extended radius intersects with the tangent.

Well, at this particular point, the radius is the extended line.

So point P and point Q are equal here and they are the point 1, 0.

So cosine theta is the X coordinate, which is 1 and sine theta is the Y coordinate, which is 0 and tan theta is the Y coordinate which is also 0.

Now if we look at the 90 degree column, again using the unit circle, we can see that when the angle is 90 degrees, so when the angle of rotation in an anti-clockwise direction from the positive X axis is 90 degrees, we've got the radius vertical.

So sine theta is equal to 1 because the point P on that circumference is now the coordinates 0, 1.

So sine theta is equal to 1 cosine theta is equal to 0 because that's the X coordinate and tan theta is undefined.

And that's because there wouldn't be a point Q.

That extended line will never intersect with the tangent because they are parallel.

So tan of 90 degrees is undefined because there is no point of intersection.

So we're now up to the second learning cycle where we're gonna be looking at spotting relationships.

So do these exact values that we've just seen it only hold if the right-angled triangle has the sides of these exact lengths.

Another way of thinking about this is, can we only deduce them from a right-angled isosceles triangle where the equal edges are 1 unit? I'd like you to pause the video and think about that question before we talk about it a bit further.

So the answer to the question is no.

And similar triangles is one way of explaining this, that regardless the size of the triangle, they will continually be in that same proportion.

So let's have a look at similar triangles a little bit further to investigate these relationships.

So on the left hand side we've got a right-angled isosceles triangle where the equal edges are 1 unit long, the hypotenuse is root 2.

In this case we call that 1 unit that's marked the adjacent because it is between the 45 degree angle and the 90 degrees and we know that the root 2 is the hypotenuse.

The second right-angled triangle on the screen is also a right-angled isosceles triangle and it is similar to the 1 we have on the left.

The only difference is its size.

So if we look at the 2 adjacent edges, our 1 unit triangle needs to be multiplied by 4 root 2 in order to become 4 root 2 centimetres where our units are in centimetres.

So we could say that the linear scale factor between these 2 triangles is 4 root 2, and when we enlarge shape, all lengths need to be multiplied by the same linear scale factor.

So if we wanted to calculate X, then we need to multiply our hypotenuse of root 2 by the same linear scale factor, which is 4 root 2.

So what's root 2 times 4 root 2? Well, it's 8.

So we now know that the X, the hypotonus on this similar triangle would be 8.

But where does the exact trigonometric values come into it? Well, if we think about the exact trigonometric value of cosine 45 degrees, cosine because we have the adjacent and the hypotenuse, well the left hand triangle tells us that the exact trigonometric value is 1 over root 2, which we know we can rationalise the denominator to be equivalent to root 2 over 2.

So if that is the ratio, if that is the exact trigonometric value for cosine 45, then that will be the same on any similar triangle.

And so cosine 45 degrees on the right hand triangle could be written as 4 root 2 over X, that is the adjacent over the hypotenuse.

But we know that that is equivalent to 1 over root 2 or root 2 over 2.

So we can set up an equation to find X and that means X would need to be 8.

So here's a quick check for you.

Is it true or false that the exact trigonometric ratios for 0 degrees, 30 degrees, 45 degrees, 60 degrees and 90 degrees must be memorised? And justify your answer.

Press pause and then when you're ready to check, press play.

So this is false.

They do not need to be memorised as you can always derive them using an equilateral triangle, an isosceles right-angled triangle and the unit circle.

You can memorise them if you want to, but there's no need to when you can just derive them whenever you need to make use of them.

So onto Task B, the final task of the lesson, on question 1, you need to find the lengths of the unknown sides without using a calculator.

So in here you've got some 45 degree angles, but you've also got some 30 degree and some 60 degree angles.

So pause the video and then when you press play, we'll move on to the last 2 questions of the task.

So here we've got question 2 and question 3 on this slide.

On question 2, you need to prove that the angle is 30 degrees within that triangle.

And on question 3, you need to find the exact value of a sum without using the calculator.

So pause the video whilst you work on question 2 and question 3.

When you're ready to go through the answers to Task B, press play and that's what we'll do.

So here we're up to question 1.

We need to find the lengths of the unknown sides and without using the calculator.

So on question 1 part a, we've got 45 degree angle and that is 1 of our exact trigonometric values.

We've got the opposite as a and the adjacent as 1.

So if we're working with the opposite and the adjacent, then it's the tangent ratio.

Tan or 45 degrees is equal to 1.

So a over 1 has to equal 1.

Well, what value of a must that be? Well, that must be 1.

If we now look at the part b, this time our angle is 60 degrees, our opposite is B and our adjacent is 10.

So we're still working with the tangent ratio.

Tan of 60 degrees, well, what's tan of 60 degrees? This is 1 that you would have to derive from an equilateral triangle.

And so we're gonna look at tan of 60 degrees, which is root 3 as an exact value, and we'll say that tan of 60 degrees is equal to b over 10 and 10 of 60 degrees is also equal to root 3, which implies that b over 10 is equal to root 3 and therefore, b has to be 10 root 3.

On part C, our angle is 30 degrees.

It's the opposite that we're trying to calculate the length of and we have the adjacent again.

So it's still the tangent ratio, but this time 10 of 30 degrees is equal to c over 10, the opposite over the adjacent, and we also know that the exact trigonometric value of 10 of 30 is root 3 over 3, which means that c over 10 is equal to root 3 over 3, hence c is 10, root 3 over 3.

So we are equating the exact tri geometric value with our ratio of our opposite and our adjacent in this case.

Moving on to part D, d is the hypotenuse and the 10 centimetre edge is our opposite.

Our angle is 45 degrees.

So what is sine of 45 degrees? Well sine of 45 degrees is root 2 over 2.

Sine of 45 degrees in this triangle is 10 over d.

So we are going to say that 10 over d is equal to root 2 over 2.

So with some manipulation of surds, we can get an exact value for d as 10 root 2.

Moving on to E, we've got an angle of 45 degrees.

It's the adjacent that we're trying to calculate and we have the hypotenuse.

So it's cosine that we'll be using.

Cosine of 45 degrees is equal to root 2 over 2.

That's the exact trigonometric value.

But in this triangle cosine 45 degrees is equal to e over 46.

So e over 46 is equal to root 2 over 2.

Once again, doing some manipulation of surds, we can say that e is 23 root 2.

Finally we're onto part F.

f is the hypotenuse, opposite is 46 and the angle is 60 degrees.

So this is sine of 60 degrees, which is root 3 over 2.

And so root 3 over 2 is equal to 46 over f, which means the f is 92 root 3 over 3 as an exact form.

If we now look at question 2 and question 3, question 2 prove that a is equal to 30.

Well, here we have the opposite and we have the hypotenuse.

So that means we can look at using the sine formula and the sine ratio.

So sine of a is equal to the opposite divided by the hypotenuse, which simplifies to 1/2.

And therefore, a is 30 degrees because when is sine equal to 1/2? When a is equal to 30.

And on to question 3, find the exact value of sine of 60 degrees plus cosine of 45 degrees.

So sine of 60 degrees using our equilateral triangle, we can derive that value.

So sine of 60 degrees is root 3 over 2.

Cosine of 45, so using our right-angle isosceles triangle, we can derive that, is root 2 over 2.

So if we find the sum, that's root 3 plus root 2 all over 2.

Well done if you managed that.

So, to summarise today's lesson, there are exact values for the trigonometric ratios of 0, 45 and 90 and they can be derived from an isosceles triangle where 2 side lengths of 1 unit and a 90 degree angle between them.

Well done today and I look forward to working with you again in the future.