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Hello, my name is Dr.

Rowlandson, and I'm thrilled that you're joining me in today's lesson.

Let's get started.

Welcome to today's lesson from the unit of trigonometry.

This lesson is called the unit circle.

And by the end of today's lesson, we'll appreciate that the trigonometric functions are derived from measurements within a unit circle.

This lesson will introduce some new keywords.

These words will be explained in much more detail during today's lesson.

But if you'd like to read these words before we get started, feel free to pause the video and do so.

Here are also some previous keywords that'll be useful during today's lesson.

If you'd like to remind yourself what these words mean, feel free to pause a video and then press play when you're ready to continue.

This lesson has two learning cycles.

In the first learning cycle, we'll be introducing the concepts of sine and cosine.

And then the second learning cycle, we'll introduce tangents.

Let's start with an introduction to sine and cosine.

Let's begin by solving this problem before we get started.

How far is the point with coordinates 4, 3 from the origin? The origin is the point with coordinates 0, 0.

We can find this distance using Pythagoras's theorem.

Pause the video while you have a go with that, and press play when you're ready to work through it together.

We can use Pythagoras's theorem here by first drawing a right-angled triangle like so.

What we are trying to find is the distance between those two coordinates.

If we call that x, then that's the hypotenuse on our right-angled triangle.

So x squared is equal to four squared, which is the horizontal distance squared plus three squared, which is the vertical distance squared.

And then we have x squared equals four squared plus three squared.

We can simplify and solve for the value of x to get five units.

So what we have here is a right-angled triangle where the length of the hypotenuse is five units, and it has one of its vertices at the origin.

Izzy creates a similar triangle on a coordinate grid with a hypotenuse with length of one unit.

What are the coordinates of point P? Well, we have the length of the hypotenuse in each triangle, so we can use that to get a scale factor from the left-hand triangle to the right-hand triangle of 1/5.

We can then apply that scale factor to the length of the horizontal edge on these triangles.

So that's 1/5 of four to get 0.

8 on the right-hand triangle.

We can do the same with the vertical length, 1/5 of three, which we'll get 0.

6 as the vertical height of that right-hand triangle.

So the coordinates of point P are 0.

8 and 0.

So let's take a look at Izzy's triangle in a bit more detail.

The triangle below has a hypotenuse of length one unit with one of its vertices at the origin.

Alex says, "What would happen if we kept the hypotenuse at a fixed length of one unit and change the position of point P?" Hmm, what do you think would happen in that situation? Pause the video while you think about it, and press play when you're ready to continue.

Well, if you'd like to explore this for yourself, this link in the slide deck takes you to a GeoGebra file where you can move point P for yourself and see what happens.

If it did so, it would look something a bit like this.

So back to our problem here, Izzy says, "Point P would move in a circle around the origin, and the radius of the circle would be one unit." And this is what the unit circle is.

The unit circle is a circle on a coordinate grid that is centred at the origin and has a radius of one unit.

That means the distance from the centre of this circle, which is at the origin, to every point on its circumference, is one unit.

We can see that on the axes in this way.

You can also see it in other parts of the circle as well.

So if we have a unit circle like this with its radius marked, which has length one unit, a right-angled triangle can be made between the radius and the x-axis, like so.

Lucas says, "The hypotenuse of the triangle is the radius of the circle.

So the hypotenuse has length of one unit." The two other lengths on the triangle can be found from the coordinates of the point where the radius intersect a circle.

So here, we can see the radius intersects a circle at the point with coordinates 0.

8 and 0.

So that means the horizontal side in this triangle has length 0.

8 units, and this can be deduced from the x-coordinates.

And the vertical side in this triangle has length 0.

6, and this could be deduced from the y-coordinate.

As the radius is rotated around the circle, we can consider how angle of rotation affects the lengths on this triangle.

So here, the radius is horizontal, and we can't actually see a triangle at this point, but if we rotate that radius around the origin like so, we can then see a triangle appear.

And let's look at the length on this triangle as I continue to rotate the radius around the origin.

What happens? So focusing on the lengths of the triangle, what changes as I rotate the radius around the origin and what stays the same? Pause the video while you think about these questions, and then press play when you're ready to continue.

Well, as I rotate the radius around the origin, the length of the horizontal and vertical edges of that right-angled triangle vary.

And what stays the same? Well, the length of the hypotenuse is always one unit.

So let's check what we've learned.

The unit circle is centred at the origin and has a radius of length, what goes in that blank there? Pause video while you write it down, and press play when you're ready for an answer.

The unit circle is centred at the origin and has radius of length one unit.

Here we have the unit circle with a right-angled triangle inside it, and we can see that the angle of rotation between the x-axis and the radius so far is 37 degrees.

Which of the lengths of the triangle inside the circle will vary as the size of the angle changes? Is it a, the horizontal length, b, the hypotenuse, or c, the vertical length, or is it multiple of these answers? Pause video while you make your choices, and press play when you're ready for an answer.

The answers are a, the horizontal length and c, the vertical length.

They will change as the angle changes.

So we've been looking at how the lengths of the triangle change as the angle changes in this unit circle.

Let's focus now on the coordinates.

Here we can see the unit circle and the point where the radius meets the circumference has the coordinates 0.

94 and 0.

34.

As the radius is rotated around the circle, the angle of rotation affects the y-coordinate of the point on the circle.

So let's now introduce the word sine to describe what's going on here.

The sine of an angle is the y-coordinate of the point where the radius has been rotated through that angle.

So, for example, sine of 20 degrees is 0.

34 because we can see when the angle of rotation here between the x-axis and the radius is 20 degrees, that point has a y-coordinate of 0.

34.

Now the angle of rotation is 37 degrees.

And we can see the point has coordinate 0.

8, 0.

The y-coordinate is 0.

Therefore, the sine of 37 degrees is 0.

And here, sine of 67 degrees is 0.

92.

If you have access to the slide deck, at the bottom of this slide contains a link to an interactive GeoGebra version of what we've just seen here.

Feel free to use that to explore this for yourself.

If you do so, what we've just seen will look something a bit like this.

Now, so far, we've just seen examples within the first quadrant of the coordinate grid, but we can also consider what happens when the radius is rotated beyond the first quadrant.

For example, here, the radius has been rotated 127 degrees.

Now the y quadrant there is 0.

8, so the sine of 127 degrees is 0.

Sophia says, "There is still a right-angled triangle, but the angle of rotation is no longer one of its interior angles." As we continue to rotate the point even further around the circle, we can see now we have a point with coordinate minus 0.

6 and minus 0.

8 when the angle of rotation is 233 degrees.

So the sine of 233 degrees is minus 0.

Sophia says, "There is still a right-angled triangle, but triangles can't have negative lengths.

So in this quadrant, the sine of the angle is not equal to the height of the triangle," which is what it was in the previous examples we've seen.

You can explore this further for yourselves by using that GeoGebra link.

And if you did so, it would look something a bit this.

As the radius is rotated around the circle, the angle of rotation also affects the x-coordinate of the point on the circle.

So, for example here, we can see when the angle of rotation is 20 degrees, the point has coordinates, 0.

94, 0.

34.

So the x-coordinate is 0.

94.

So let's now introduce the word cosine to describe what's going on here.

The cosine of a given angle is the x-coordinate of the point where the radius has been rotated to form that angle.

So here, we can see when the linear angle is 20 degrees, the point has coordinates, 0.

94, 0.

34.

The x-coordinate was 0.

94, so the cosine of 20 degrees is 0.

94.

Here, the angle of rotation is 37 degrees.

We can see the x-coordinate is 0.

8 there.

So the cosine of 37 degrees is 0.

And the cosine of 67 degrees is 0.

If we do what we've just done again with the interactive version on GeoGebra, it would look something a bit like this.

We could also consider what happens to cosine as the radius is rotated beyond the first quadrant.

So here you can see the angle is 127 degrees.

The point has an x-coordinate of minus 0.

So the cosine at 127 degrees is minus 0.

Sophia says, "There is still a right-angled triangle, but the angle of rotation is no longer one of the interior angles." She also says that "Also, triangles can't have negative lengths.

So in this quadrant, the cosine of the angle is not equal to the base of the triangle." So let's check what we've learned there.

The blank of an angle is the y-coordinate of the point where the radius has been rotated through that angle.

What goes in that blank? Pause video while you write the word down, and press play when you're ready for an answer.

The answer is sine.

The cosine of an angle is the blank of the point where the radius has been rotated through the angle.

What goes in that blank? Pause the video while you write it down, and press play when you're ready for an answer.

The answer is x-coordinate.

So let's look at the diagram here.

The sine of 52 degrees is blank.

What goes in that blank? Pause video while you write it down, and press play when you're ready for an answer.

Well, sine is the y-coordinate here, which is 0.

79.

The cosine of 150 degrees is blank.

What goes in that blank? Pause the video while you write it down, and press play when you're ready for an answer.

The answer is minus 0.

87.

Let's now talk notation.

The angle can be represented by the Greek letter theta.

That's how you write it, and that's how you say it.

In the unit circle, theta is the angle that the radius is rotated anti-clockwise from its starting point at 0, 1.

So it's this angle here.

And what about notation when it comes to sine and cosine? Well, a short way to write the sentence sine of the angle is sine theta, where S-I-N means sine, and theta is the angle, which we put in brackets after the letters S-I-N.

A short way to write the sentence cosine of the angle is cos theta, where cos means cosine, and theta is the angle which we put in the brackets after the letters C-O-S.

So let's now think about how sine and cosine relate to a right-angled triangle that is inside a unit circle.

Because triangles cannot have negative lengths, therefore, for triangles in a unit circle, sine theta only equals the vertical length when its value is positive.

And cos theta only equals the horizontal length when its value is positive.

So with that in mind, in which quadrants do both sine theta and cos theta take positive values? Pause the video while you think about this, and press play when you're ready to continue.

So let's take a look at these quadrants one at a time.

In the first quadrant, we can see that both sine and cosine are positive.

Therefore, the vertical and horizontal lengths of that triangle will be equal to the values of sine and cosine.

In the second quadrant, well, the value of sine theta would be positive but not the value of cos theta.

So the horizontal length of the triangle would not be equal to cos theta.

In the third quadrant, both sine and cosine are negative.

Therefore, neither of those will be equal to lengths on the triangle.

And in the fourth quadrant, we can see that cosine is positive, but sine is negative.

Therefore, the vertical length of that triangle will not be equal to the value of sine theta.

So that means the first quadrant is the only quadrant where the values of sine and cosine are equal to lengths on the triangle.

So in that case, we only really need to think about the first quadrant of this unit circle when we're thinking about lengths on a right-angled triangle.

The diagram here shows the sector of the unit circle that lies in this first quadrant.

The points around the circumference show the angle of rotation, theta from the horizontal axis to the radius at each point.

So, for example, if I draw the radius here, we can see that the angle between the horizontal axis and the radius is 20 degrees.

If you'd like to access an interactive version of this tool, you can click on the link on the slide to take you to it.

So how could we use this to help us? Well, we could use this quarter of a circle to read off values of sine and cosine.

So, for example, with the right-angled triangle I've drawn here, we can read off the value of sine by looking across to the y-axis and seeing that sine of 20 degrees is 0.

34.

Now sin of 20 degrees does have more digits after those first two decimal places, but this is as about as accurate as we can read it from this graph here.

Here, we can read off the value of cos 20, which is 0.

94.

Once again, there are more digits after those first two decimal places, but this is about as accurate as we can read it on this piece of graph paper.

If you'd like to explore this using the interactive tool, feel free to click on the link, and it looks something a bit like this.

We have a unit circle centred at the origin, and to find the angle of rotation for the radius, we can either click to display a protractor or click here to display a more accurate reading of the angle.

To explore sine, we can either show the lengths of the triangle or show the y-coordinate at the point.

And if we move that point around, we can see how these things change.

To explore cosine, we need to untick those, and we can either click the lengths or cosine for the cosine function and move the point around again.

So let's check what we've learned.

What is the value of theta? Pause video while you write it down and press play when you're ready for.

Theta is 35.

The angle is 35 degrees.

So here, the angle has been set up 40 degrees, and the x value and y value of that point has been marked.

So what is sine of 40 degrees? Pause the video why you write it down, and press play when you're ready for an answer.

The answer is 0.

64.

What is cosine of 65 degrees? Pause video, write it down and press play when you're ready for an answer.

The answer is 0.

42.

Okay, it's over to you now for task A.

This task contains two questions, and here is question one.

The diagram shows the sector of the unit circle that lies in the first quadrant.

You need to use that diagram to complete the table.

And then use your table to answer parts b, c and d.

Now, you can either do this using the version of the quarter circle you can see on the screen here or if you have access to that link on the previous slides, you can use that to read them off a little bit more accurately, but it doesn't matter which one you use.

So pause the video while you have a go at it, and press play when you're ready for question two.

And here is question two.

It's all about doing the same thing again, but this time, reading off the values of cos theta.

Pause the video while you do this, and press play when you're ready for some answers.

Okay, let's take a look at some answers now.

Part a, here are your values of sine theta.

Now one thing to bear in mind that your answers may be slightly different to these ones depending on how accurately you were able to read it from the graph, but hopefully you've got answers that are pretty close to these.

And part b, what happens to the value of sine theta as the angle increases in this quadrant? It increases as well.

Its lowest value is zero, and its greatest value is one.

And I need to use your table to find the angle where sine theta is 0.

So looking at our table, we can see sine theta is 0.

5 when it's 30 degrees.

Using the diagram, we need to find the angle where sine theta is 0.

That's 53 degrees.

Then question two, when we fill our table, our values of cos theta are these.

Now once again, your answers might be a little bit different to these depending on how accurately you are able to read it off the graph.

Part b, what happens to the value of cos theta as the angle increases in this quadrant? Well, it decreases.

Its lowest value is zero, and its greatest value is one, but it decreases from one to zero.

We need to find the angle where cos 30 is 0.

5, that's 60 degrees.

And then find a value of theta where cos theta equals sine 40 degrees.

Well, we can read it off the graph, or we can look back at our previous table from question one if we have that to hand.

Either way, 50 degrees.

And then find the angle where cos theta equals sine theta.

Once again we can do it either by comparing the tables from question one and two or using the graph and looking for where they are the same.

That's at 45 degrees.

Okay, that's sine and cosine.

Now let's look at tangent.

Let's start by defining what a tangent is to a circle.

A tangent of a circle is a line that intersects a circle exactly once.

So like we can see here, this line just about touches that circle exactly once.

Here are some more examples of tangents, and here are some examples that are not tangents.

The one on the left that intersects a circle twice, the one on the right doesn't intersect it at all.

So let's now apply what we've just learned about tangents to a unit circle.

The diagram here shows the unit circle with a tangent at x equals one.

Sam says, "If we draw a line through the origin, it'll intersect the tangent exactly once," like so.

Andeep says, "This creates another right-angled triangle between the line, the tangent, and the x-axis." Can you see where that right-angled triangle is? Here it is.

So here we have the line set at a horizontal position, and we can see intersects a tangent at the 0.

10.

Rotating this line affects where it intersects a tangent.

So let's think again about what varying the angle does to the length on the triangle and also the coordinates as well of that point.

What changes as the angle varies, and what is constant as the angle varies? Pause video while you think about this, and impress play when you're ready to continue.

So what changes as the angle varies? Well, the y-coordinate changes and so does the length of the side and the triangle that is opposite the angle.

The hypotenuse also changes as well.

And what is constant as the angle varies? Well, the x-coordinate is constant, that is always one.

Also the length of the side on the triangle that is adjacent to both the angle of rotation and the right angle, that is also constant.

That horizontal length is always one as well.

Now earlier, we defined what a tangent of a circle is, but the word tangent has multiple uses.

The tangent of an angle is the y-coordinate of the point where the line, the triangle's hypotenuse in this case, intersects the tangent line in this unit circle.

So here, we can see that when the angle is 20 degrees, that hypotenuse intersects a tangent at the y-coordinate of 0.

36.

Here, we can see that when the angle is 37 degrees, the hypotenuse meets a tangent with the y-coordinate of 0.

75.

So the tangent of 37 degrees is 0.

75.

And here we can see the tangent of 45 degrees is one because the hypotenuse has met a tangent with the y-coordinate of one.

So let's talk about notation again.

A short way to write tangent of the angle is tan theta, where tan is an abbreviation of the word tangent, and theta is our angle of rotation, and we write that angle inside the brackets.

For example, tan 45 degrees equals one.

Aisha says, "I wonder what happens to the value of tan theta when the angle increases beyond 45 degrees." Well, if we continue to rotate that hypotenuse, it will intersect the tangent at a higher point, so we won't be able to see it on the graph as it currently is.

But zooming out allows to see what happens to the value of tan theta when the angle of rotation is increased beyond 45 degrees.

It looks a bit like this.

Here we have 10 of 55 degrees.

It's 1.

43.

Aisha says, "The length of the side that is opposite the angle is now longer than the radius of the circle.

That means that tan theta can take values which are greater than one." Here, 10 of 63 degrees is equal to two.

Aisha says, "It looks like as the angle keeps increasing, the value of tan theta also keeps increasing.

I wonder how long it'll keep increasing for." Hmm, what do you think to that? Well, when the angle is 90 degrees, it doesn't intersect the tangent at all.

So a triangle can't be formed.

And here we can see we've rotated it 117 degrees, and it's now intersecting somewhere with a negative y-coordinate.

Aisha says, "When the line is rotated slightly beyond 90 degrees, the line intersects a tangent at a point with a negative y-coordinate." She also says, "The angle of rotation is no longer an interior angle of the triangle here.

Also, triangles can't have negative lengths." If you have access to this slide deck, you can click on the link at the bottom of this slide to go to an interactive GeoGebra version of what we've just done.

And if you do that, it looks something a bit like this.

So once again, it looks like we can just use quarter of the circle here to create some right-angled triangles.

The diagram shows the sector of the unit circle that lies in the first quadrant.

By drawing a tangent at x equals one, we can find the tangent of some angles in the unit circle.

For example like this, and we can use it to find tan of 40 degrees.

We can draw a line through the origin and through the 40 degree marker on that arc, see where it intersects the tangent, and then look at the y-coordinates at that point, which is 0.

84.

And if you have access to the slide deck, you can click on the link to access an interactive version of this tool, and it looks something a bit like this.

We have our unit circle centred at the origin, and to find your angle of rotation, you can either click Protractor or click Angle.

To draw a tangent, click Tangent, and then here display the lengths or the value of the tangent function.

To see more, you can scroll up or you can zoom out.

And then move the point around a circle and see how things change.

So let's check what we've learned.

What is the value of tan 50 degrees? You've got three options to choose from.

Pause video while you make a choice, and press play When you're ready for an answer.

The answer is a, 1.

19.

What is the value of sine 50 degrees? Got three options again.

Pause video while you choose, and press play when you're ready for an answer.

The answer is b, 0.

77.

What is the value of cos 50? Pause the video while you make a choice, and press play when you're ready for an answer again.

The answer is c, 0.

64.

Okay, it's over to you now for task B.

This task contains two questions, and here is question one.

The diagram shows the first quadrant of the unit circle.

You need to first draw a tangent at x equals one, and then use it to answer questions b, c, d and e.

And if you want to access the interactive version of this instead, feel free to click on the link.

Pause the video while you do it, and press play when you're ready for question two.

Here is question two.

It's all about putting together everything we've learned in today's lesson.

Use this diagram or your previous answers to fill in the table, and then answer the question in part b.

Pause the video while you do this, and press play when you're ready for some answers.

Okay, let's see how we got on with that.

When we draw a tangent, x equals one, it should look like this.

We can then use it to find the value of tan 35 degrees, which is 0.

7, tan 55 degrees, which is 1.

And we can use the diagram to find the angle where tan theta equals one by going to one on the vertical axis, going across to the tangent, and then diagonally to the origin and seeing what the angle was.

It's 45 degrees.

And we can do the same thing to find the angle where tan theta equals 0.

5, and it's 27 degrees.

Now once again, you might have slightly different answers to these depending on how accurately you are able to read off the graph, but hopefully, you've got something close to these values.

And with question two, once we fill in the table, we should get these values, which again, may differ depending on how accurate you could read them off the graph.

And for each angle, divide the value of sine theta by the value of cos theta and see what you notice.

When you do so, you should get answers that are approximately equal to the value of tan theta.

Now, they might not be exactly equal to tan theta based on the values you've got because they might not be read completely accurately.

Even these values on the table here, they're only given to maximum of two decimal places.

And actually, for most of these values, there are more digits after the second decimal place.

So even these are not quite exact.

But if they were exactly accurate, when you divide sine theta by cos theta, you would get tan theta.

Fantastic work today.

Let's now summarise what we've learned in this lesson.

The unit circle is a circle of radius of one unit, and the unit circle is centred at the origin, and we introduced the words sine, cosine, and tangent today, which are our trigonometric functions.

The sine of an angle is the y-coordinate of the point where the radius is being rotated through the angle, and the cosine of an angle is the x-coordinate of that point.

And the tangent function of an angle is the y-coordinate of the point where the extended radius intersects the tangent line to the unit circle.

That's x equals one.

That's some pretty tough going today, so great work, and thank you very much.

I've finished the video