Lesson video

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Hello, I'm Mrs. Lashley, and I'm going to work with you as we go through the lesson today.

I really hope you're looking forward to the lesson.

You're willing to give it your best shot, and I will be there to support you as we get through it.

Our learning outcome today is to be able to see how the trigonometric functions are derived from the measurements from a unit circle, and how this can be utilised.

So on the screen there are a few definitions and a diagram that you may wish to pause the video and re-familiarize yourself with.

You will have met them before, but they are a key part to this lesson.

Press play when you're ready to move on.

So in this lesson on checking and securing understanding of the unit circle, we're gonna break it into two parts.

The first learning cycle is revisiting sine and cosine.

And then the second learning cycle will be revisiting tangent.

So let's make a start at revisiting the sine and cosine trigonometric ratios.

So let's first go back over what the unit circle is.

The unit circle is a circle on a coordinate grid.

So we can see a circle there on a grid, and we've got the x and y marked.

More importantly, this unit circle is centred at the origin and the origin we know is the coordinates zero, zero, and its radius is one unit.

So the coordinates at where it crosses these coordinate axes is one, zero, zero, one, negative one, zero and zero, negative one, all of which are one unit from the origin.

So that's our radius.

However, there are all the other points along the circumference that are also one unit from the origin.

So any of these four points.

They're on the circumference, they're on the unit circle.

They are one unit from the origin, the centre.

So a right-angled triangle can be made between the radius and the x-axis.

So if we look here, we've got a point on the circumference and where's the right-angled triangle? Well, here.

So between the radius, which is a radius of one, because it's the unit circle, and the x-axis, which we know is our horizontal axis.

So Lucas says, "The hypotenuse of the triangle is the radius of the circle." So hypotenuse is that longest edge, always opposite the right angle.

And the hypotenuse has a length of one unit, because this is the unit circle.

The two other lengths on the triangle could be found from the coordinates of the point where the radius intersects the circle.

So the point at which the radius meets the circumference.

Here we have a (0.

8, 0.

6).

Lucas says that the length of the triangle's horizontal edge can be deduced from the x-coordinate.

So we can see that it's 0.

8 and that's the x-coordinate.

And the length of the triangle's vertical edge can be deduced from the y-coordinate using the y-axis to get that length.

As the radius is rotated around the circle, so remember a radius doesn't have to be in one particular direction, we can consider how the angle of rotation affects the lengths on the triangle.

So currently, there is no rotation, it's on that x-axis.

And remember, the angle is between the radius and the x-axis.

So our angle here would be zero.

But if we rotate it to 20 degrees, then we've got our hypotenuse is equal to one because that's the radius of the circle.

And we can see the point on the circumference.

If we rotate it further to 37 degrees, what has changed? What hasn't changed? Or here, to 67 degrees.

So what changes and what stays the same? Pause the video, it might be that you need to rewind to go back and look at what changes and what stays the same when we rotate the radius.

Press play when you're ready to go through that.

So what's changing? Well, the lengths of the horizontal and vertical edges vary.

So what's marked as blue and green on the triangle, the horizontal edge and the vertical edge, the two that are perpendicular to each other, they have varied, they change as the angle of rotation changes.

But what stays the same? Well, the length of the hypotenuse.

So the length of the hypotenuse is fixed at one unit, because it is the unit circle.

So here's a quick check on that.

The unit circle is centred at the origin and has a radius of length? You need to fill in that blank.

Press pause whilst you think about it.

When you're ready to check, press play.

So a length of one unit.

Another check.

Which lengths or length of the triangle inside the unit circle will not vary as the size of the angle changes? So A, the horizontal length.

B, the hypotenuse, or C, the vertical length? Pause the video and when you're ready to check your answer, press play.

The hypotenuse is the thing that does not vary.

So the hypotenuse is fixed as one, whereas the horizontal and vertical lengths do change as the angle of rotation changes.

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

So here again, we've got our radius 20 degrees, a turn of 20 degrees from the x-axis.

And we can see that the y-coordinate is 0.

34 when the angle of rotation is 20 degrees.

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

So our trigonometric ratio sine on when we're using a unit circle, is the vertical height.

And in this particular example it's 0.

34.

So the sine of 20 degrees is 0.

34, it's our y-coordinate.

There is a link to an interactive version for both sine and cosine, which we're covering in this learning cycle, where you can have a look at the lengths as the angle changes.

So here's an another example.

The sine of 37 degrees is 0.

So the angle is now 37 degrees, and our sine value is 0.

And here when the sine of 67 degrees is 0.

92, it's the y-coordinate of the point that the radius meets at the circumference.

We can also consider what happens when the radius is rotated beyond the first quadrant.

So here we've got a triangle in the third quadrant.

The sine of that angle, 233 degrees, so a reflex angle, is negative 0.

The y-coordinate of that point is negative 0.

Sofia says, "There is still a right-angled triangle, and you can see it on the diagram, but triangles cannot have negative lengths.

So in this quadrant, the sine of the angle is not equal to the height of the triangle." So it's not equal to the height of the triangle because that height would be 0.

8, it would be the modulus of negative 0.

So now if we consider the x-coordinate, so we considered the y-coordinate, which is the sine value.

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

So in this triangle here, cosine of 67 degrees is 0.

39.

But we can also consider what happens when the radius is rotated again outside of the first quadrant.

So here we've got a triangle outside into the second quadrant.

So the cosine of 127 degrees, remember that is the angle of rotation from the x-axis, the positive x-axis to the radius is negative 0.

Its x-coordinate is negative 0.

And Sofia says, "There's still a right-angled triangle, but the angle of rotation is no longer one of its interior angles." So, previously, we had an angle inside of the triangle.

But 127 degrees is the exterior angle at the origin.

So here's a check for you.

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

Pause the video if you need to go back and re-watch anything, then do.

And then when you're ready to check your answer, press play.

So the sine of an angle gives you the y-coordinate.

Next check.

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

Once again, pause the video, if you need to go back and re-familiarize yourself, do so.

And then when you wanna check and move on, press play.

So the cosine of an angle is the x-coordinate of the point where the radius has been rotated through that angle.

So using the diagram, what's the answer? The sine of 52 degrees is? So pause the video and when you're ready to check, press play.

So it's 0.

79 because remember, it's the y-coordinate.

The cosine of 150 degrees is? Pause the video.

When you're ready to check, press play.

Negative 0.

87.

It's the x-coordinate.

So triangles cannot have negative lengths.

Sofia mentioned that earlier.

Therefore for triangles in a unit circle, sine theta only equals the vertical length when its value is positive.

Because otherwise, it doesn't make sense.

So also, cosine theta only equals the horizontal length when its value is positive.

Because once again, we cannot have negative lengths.

So if cosine theta is a negative value, then it doesn't represent the horizontal length of a triangle.

So in which quadrants do both sine theta and cosine theta take positive values? So you've got the first quadrant, second quadrant, the third quadrant, and the fourth quadrant.

So pause the video and have a think about that.

When do both sine theta and cosine theta have positive values? Press play when you're ready to move on.

So in the first quadrant, they both have positive values.

In the second quadrant, do they both have positive values? No.

Cosine has a negative value, but sine has a positive value.

What about in the third quadrant? Do they both have positive values? No, they both have negative values.

The x-coordinate is negative, and the y-coordinate is also negative.

And finally, what about the fourth quadrant? Do any of these have positive values? Do they both positive values? Well they don't both have positive values, but they, one of them is positive and that's cosine.

So cosine theta is positive in the fourth quadrant, but sine theta is not.

The y-coordinate is negative.

So the answer to which ones have both positive values is the first quadrant only.

So now, we're gonna look at the unit circle, but only a sector of it.

And we're keeping to the first quadrant.

So this diagram shows you the sector of the unit circle in the first quadrant.

The points around the circumference show the angle of rotation.

So that is the angle, or our theta value.

From the horizontal axis to the radius at each point.

So here if we've got this radius drawn and it meets the 20 degrees on the circumference, then that's because the angle of rotation is 20 degrees.

There is a link to a file that is interactive for you to have a go and look and get more familiar with this diagram if you wish to.

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

So we've met that already.

The sine of an angle is the y-coordinate.

So for this one here, we can read that y-value, and it's 0.

34.

So sine of 20 degrees is 0.

34.

The cosine of the angle is the x-coordinate of the point where the radius is, having been rotated through the given angle.

So cosine of 20 degrees is 0.

94.

So we're using our x-coordinate.

So here's a quick check.

What is the value of theta in this diagram? Press pause and when you're ready to check, press play.

30 degrees.

So theta is 30, because the radius is meeting the point on the circumference marked as 30 degrees.

Using the diagram, what is sine of 60 degrees? Pause the video and when you're ready to check it, press play.

So you should have been taking the y-coordinate, and it's 0.

87.

So sine theta is our vertical axis.

And so we're taking sine of 60 degrees is 0.

87.

Another check.

What is cosine of 60 degrees? Pause the video.

When you're ready to check your answer, press play.

So cosine 60 degrees is a half.

So this time it's the x-coordinate.

And you can see that the x-axis is labelled as cosine theta.

Remember that theta is that angle of rotation from the positive x-axis to the radius.

So we're onto the first task of the lesson, which is revisiting sine and cosine.

So on question one, you're gonna make use of that diagram that you've just been working with.

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

So using the diagram, estimate and then calculate the exact value of sine theta.

So your estimate is using the diagram and then you're going to use the calculator to get the exact value to two decimal places.

So pause the video and then when you press play, we'll move to the next question.

So here's question two, another similar question to question one, but this time, it is cosine.

So cosine 10 degrees, cosine of 30 degrees, et cetera.

So use the diagram to get your estimate and then your calculator and round your answer to two decimal places.

Press pause whilst you're doing that, when you press play, we'll go through our answers to task A.

So here are my answers to question one.

So on that first column that says my estimate, you're going to be approximately 0.

So if you've put a value that's close to 0.

2, then that's absolutely fine.

There is always going to be some inaccuracy when we're using this diagram.

And then your calculator value should be 0.

17, to two decimal places.

So all our calculators should agree.

The only reason that we may have got different answers in that calculator value column is if your calculator was in radians, and not in degrees.

So check to see that you are in the degree mode.

Most calculators will have a D on the screen to indicate they're in degrees.

If you have an R, then you're in radians, and that needs to change.

Here's the answers to question two.

Again, it was a very similar task, but this time it was for cosine.

And again you should see that your estimate needs to be in the area of my estimate, but we may not have exactly the same answers.

So now we're up to the second learning cycle, where we're gonna revisit the third trigonometric ratio of tangent.

So tangent.

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

And we've got a diagram there.

So we've got our circle and then we have our tangent.

Here are two further examples of diagrams that show tangents.

So the tangent does not need to be vertical, it does not need to be horizontal.

But it needs to only intersect the circle once.

And here are some not tangents.

Some examples of diagrams that are not showing tangents.

Yes there is a line, but they do not meet the definition to be a tangent.

So the first one you can see is intersecting the circle at two different points.

And the second one is not intersecting at all.

So we need it to intersect with the circumference of the circle exactly once for it to be a tangent.

So here we've got our unit circle.

Remember the unit circle is a circle on the coordinate grid centred at the origin with a radius of one unit.

And also this time an additional part to our diagram is the line x equals one, which is a tangent to the unit circle because of the radius being one.

Sam said, "If we draw a line through the origin, it will intersect the tangent exactly once." So here we've got a line that's passing through the origin and it is intersecting our tangent at one point.

Andeep says, "This actually creates another right-angled triangle between the line, the tangent and the x-axis." So it's not the radius anymore, because the radius would stop at the circumference.

This line is passing through the origin and up to the tangent and past the tangent.

So it's not the radius, but the right angle triangle is here.

So it's along the tangent, the x-axis and our line.

Rotating the line affects where it intersects the tangent.

And there is an interactive version of this that you could go and have a look at.

So here if our rotation of our line is 20 degrees from the horizontal, it intersects our tangent at this point.

If our angle increases to 37, it intersects the tangent at this point.

If we increase it to 45 degrees, then it intersects at this point.

45 degrees in a right-angled triangle means it will be isosceles.

And so it shouldn't be too surprising that that point has the coordinates one, one.

So what changes as the angle varies? And what is constant as the angle varies? So pause the video.

And again, you may want to rewind the video and have a look at those last few examples.

Go onto the interactive version and have a play around with that to see about what changes and what stays constant.

Press play when you're ready to go through the answers.

So the y-coordinate changes as the angle varies.

The length of the side on the triangle that's opposite the angle is therefore changing.

If the y-coordinate changes, then that length is changing.

What's constant? Well, the x-coordinate is constant.

Because the x-coordinate is the point at which it meets the tangent.

And this tangent is x equals one.

The length and side of the triangle that is adjacent to both the angle of rotation and the right angle.

That length that is along the x-axis is fixed at one because that is the radius of the unit circle.

So rotating the line affects where it intersects the tangent and the tangent of an angle is the y-coordinate of the point where the line, which is the triangle's hypotenuse, intersects the tangent line.

So for this one, when the angle of rotation is 20 degrees, the tangent of 20 degrees is 0.

36, it's the y-coordinate.

What's the x-coordinate? Well the x-coordinate is one because that tangent is the line x equals one.

For this rotation we've got the tangent of 37 degrees is 0.

75.

What's the x-coordinate? The x-coordinate is one.

And for this final one, or this example here, the tangent of 45 degrees is one.

And I spoke about that earlier that we've made an isosceles triangle.

So if the horizontal edge, the adjacent between the angle of rotation and the 90 degrees is one, then the other edge that's not the hypotenuse will be one.

So a short way to say tangent of the angle is tan, the first three letters.

And we've seen that with sine and cosine.

And then in brackets, theta.

And that theta will be whatever the angle of rotation is.

So for example, this one we'd write, tan of 45 degrees equals one.

So zooming out allows us to see what happens to the value of tan theta when the angle of rotation increases beyond the 45 degrees.

So if our angle of rotation is 55 degrees, the point at which it intersects the tangent has a y-coordinate of 1.

43.

So tan of 55 degrees is 1.

43.

And 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." When the angle of rotation is 63 degrees, tan 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 will keep increasing for." So you may want to go onto the interactive version here and have a play.

You may wanna think about this.

As we continue to increase the angle of rotation, so as the 63 degree angle increases, will the value of tan theta continue to also increase? What about tan 90? So the angle of rotation has gone from 63.

It increased, increased, increased, and now is at tan of 90 degrees.

What value is tan of 90 degrees? Well Aisha says, "When the line is rotated 90 degrees, it doesn't intersect the tangent at any point." Our tangent is now parallel to our line that passes through the origin.

So a triangle cannot be formed.

Our triangle was be formed from the point of intersection between the line that passes through the origin and the tangent and the x-axis.

So if that line that passes through the origin does not intersect the tangent, then no triangle can be formed.

We can also see what happens when our angle increases past the 90.

So when our angle is increased past the 90, we're now 117 degrees, our line does intersect the tangent.

It intersects now at a negative value.

Tan of 117 degrees is equal to negative two.

Aisha says, "When the line is rotated slightly beyond 90, the line intersects the tangent at a point with a negative y-coordinate.

The angle of rotation is no longer an interior angle of the triangle and it also, triangles cannot have negative lengths." And we met that concept in the sine and cosine learning cycle.

That yes, tan of 117 degrees does have a value, but it now doesn't represent a length on a triangle.

So here we've got our sector of the unit circle that's only in the first quadrant, and our y-axis is increased past the one.

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

So here we've got our x equals one tangent, which is perpendicular to the x-xis.

Tan of 40 degrees, we would draw our line that passes through the origin and passes through the circumference of the circle at 40 degrees.

And it's the point at which it meets the tangent.

The y-coordinate of that point is our value for tan of 40 degrees, and that's 0.

84.

So here's a check for you.

What is the value of tan of 50 degrees? Pause the video and then when you're ready to check, press play.

So the answer is A, 1.

19.

So it's not the point at which it meets the circumference of the unit circle, it's the point at which it meets the tangent.

So what is the value of sine of 50 degrees? Once again, pause the video.

And when you're ready to check, press play.

So sine is the y-coordinates when the hypotenuse is equal to one, and the hypotenuse is equal to one when it's the radius.

So we're using the y-coordinate of 0.

77.

And finally, what is the value of cosine 50 degrees? Pause the video.

When you're ready to check, press play.

So this one's the final one of 0.

64.

It is the x-coordinate of the radius.

So we're up to the last task of the lesson.

Task B.

And so on question one, you've got the diagram to help you.

The diagram shows the first quadrant of the unit circle.

For part A, you need to draw a tangent at x equals one.

For part B, you need to find the value of tan 45 degrees.

For part C, you need to find the value of tan 15 degrees.

For part D, you need to use the diagram to find the angle where tan theta equals one.

So what is the angle of rotation? And for part E, use the diagram to find the angle where tan theta is equal to 0.

So pause the video, and when you're ready for the next question, press play.

So here's the final question of Task B.

On part A, you need to complete the table and use the diagram to help you with that.

And then for part B, for each angle, divide the value of sine theta by the value of cosine theta and write down what you notice.

So pause the video, and then when you're finished with question two and you press play, we'll go through our answers to questions one and two of task B.

So question one, you needed for part A to draw the tangent.

So x equals one is the vertical line passing through the x-axis at one.

For part B, you needed to find the value of tan of 45 degrees.

So you needed to draw a line passing through the origin and the point of 45 degrees on the unit circle and then read the y-coordinate.

So the y-coordinate is one.

For part C, you need to find the value of tan of 15 degrees.

So the same process.

Draw a line that passes through the origin and the 15 degrees on the circumference of the unit circle and intersects the tangent.

Write down the y-coordinate of the point of intersection.

And it's approximately 0.

3 to one decimal place.

You may have been more accurate than that and written something along the lines of 0.

27.

Part D.

Use the diagram to find the angle where tan theta equals one.

So this time, it would be 45 degrees.

And for E, use the diagram to find the angle where tan theta equals 0.

Would be 27 degrees, approximately.

Your answers may vary slightly, depending on how accurately you read from the graph.

On question two, you need to complete the table.

So sine theta is the y-coordinate when you have the radius of the unit circle meeting the circumference.

Cosine theta is the x-coordinate of the point at which the radius meets the circumference.

And tan theta is the y-coordinate of the line that passes through the origin and meets the tangent at x equals one.

So answers again may vary depending on your accuracy, but you may wanna pause the video here just to check your values.

And then for Part B, for each angle, divide the value of sine theta in your table by cosine theta, and what do you notice? When you divide sine theta by cosine theta, the result is approximately equal to the value of tan theta.

There's gonna be some inaccuracy there, but hopefully, you'll be able to see that your value of sine theta divided by cosine theta is approximately equal to the value of tan theta in your table.

So to summarise today's lesson on checking and securing understanding of the unit circle.

Well the unit circle is a circle with a radius of one unit.

It's centred on the origin.

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

The cosine of an angle is the x-coordinate at the point where the radius have been rotated through that angle.

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

And that tangent's at x equals one.

So have a look at the diagram.

That explains it all as well very neatly.

Point P and point Q.

And the distances that you can see.

Really well done today, and I look forward to working with you again in the future.

I've finished the video