Introduction to the trigonometric functions

In this video, we will give an introduction to the trigonometric functions in relation to the unit circle with Dr. Ria Symonds. The aims and objectives for this video are to know how to measure angles in radians and to be able to recognise and sketch the graphs of the cosine and sine functions using the unit circle. We will begin by referring to our radians.

So, so far, we have seen how to measure angles using degrees, and we know there are 360 degrees in a circle. When we look at the trigonometric functions, we use a alternative angle value, and that is using radians. Now, radians relates to the unit circle. So I know that 1 radian is the angle that is created in a sector of a circle– so here’s my angle theta– but where I have the relationship that the radius of that sector on the circle is exactly the same as the arc length created by the sector. So if I was to take a sector like this, the radius length is exactly the same as measuring that little arc length there.

Then that gives me exactly 1 radian inside the sector. Now, there is another relationship that tells me that this happens 2 pi times, so I can fit the radius into the circumference of a circle 2 pi times. And since I know there’s 360 degrees in a circle, that gives me a way of converting from degrees to radians, or vice versa. So there are 360 degrees is equal to 2 pi. So we could convert from one to the other. Perhaps, if we have the 360 degrees– so 180 degrees– or we could think of this as maybe a semicircle– that’s going to be the equivalent of pi.

And if I was to half it again– let’s think about 90 degrees– so maybe, when we think about a right angle triangle– that’s half again, so that’s going to be half pie, or pi over 2. And when we use radians in this manner to measure angles, ideally, we should keep our angles in terms of pi so it keeps them exact. So try not to use decimals, unless it is stated within the question or whatever you are looking at. Now, we’re going to relate the unit circle to where the sine and the cosine functions actually come from. So I’m going to draw my unit circle on my xy axes here. So I’ve got x and I’ve got y.

And if I take the radius of that circle anywhere– so I’ve drawn it here. Here’s my radius. It’s of a length 1 because it’s a unit circle. But imagine that I’ve got a little orange dot there. I could move that orange dot around my circle. The radius would be the same, but the lengths here of this triangle that I’ve created– so let’s call this x along the x-axis, and let’s call these y along the y-axis. So the lengths of that triangle would change as I move the orange dot. I’m going to show you this in a moment. Now, what I’d like is a relationship between the x- and y-values and the angle theta created by that triangle.

Now, we’ve seen or use some ratios for the angle called SOHCAHTOA. So I can take sine of the angle, cosine of the angle, tan of the angle, and use the sides of the triangle to find one of the missing things. So for example, let’s take cos of the angle first. I could take any of them, but I’m going to go with cos. So cos theta is equal to the adjacent side of the hypotenuse. So the adjacent side for my triangle is x. So the adjacent side is adjacent to the angle. And the hypotenuse is 1. So therefore, that tells me the x is equal to cos of the angle, cos theta. Let’s do the same for sine.

So sine theta is now the opposite over the hypotenuse, and my opposite side is the y– the y value. Hypotenuse is 1 again, so now y is equal to sine theta. So I could try and plot what cos theta and sine theta looks like by using these values of x and y. And the values of x and y are basically the length of the adjacent side, or the length of the opposite side. So as I move my orange dot around the circle, let’s plot the difference of, say, how the x side changes. So it’s going to get bigger and smaller as I move around that circle and the same for the sine value, y.

So I’m going to show you this on an applet. So we’ll start with the cosine angle. So for the cosine angle, we’re looking at x, because cos theta’s equal to x. So x is the blue line that you can see on that unit circle. So it’s like the adjacent side. How does it get bigger and smaller as I move the orange dot around? So let’s take a little look. That’s going to be plotted on this graph here. So are you ready? So it starts at 1, but as I move this orange dot around, that blue line is getting smaller and smaller and moving towards 0, until we get to the top of the circle there.

Then, as I come back around the other way, we’re going to start having a negative x value– because we’re going into the negative x-axis– until I get all the way around to pi. We know pi is 180 degrees. And then, if I keep going, that value’s going to get, again, smaller and smaller, going towards 0. And then, as I move back to where I started, can you see the blue line is getting bigger and bigger and bigger until it hits 1? It’ll never get bigger than 1 because the radius of this circle is 1. And here we can see that this kind of wave shape is created, and this is the cosine graph.

We should be familiar with the cosine graph, but this is actually where it comes from. This is how we know how to sketch it. And we could do the same for the sine function. So the sine function is now the y value. It’s how long the opposite side is. Let’s think about how that changes as I move the orange circle. So the orange circle moves. The length of the opposite side gets bigger. Can you see it’s getting bigger? The red line’s getting bigger and bigger and bigger, until we get to the top here, and it’s of length 1.

And then, as we come back around the other way, it gets smaller again– smaller and smaller and smaller– until it nearly disappears and goes to 0. And then we keep going round– oh, let’s just colour that in a little bit more– we keep going round, and we’re now in negative value, because we’re underneath in the y-axis, the negative axis. So it gets more and more negative towards minus 1, and then we come around, getting smaller and smaller, to 0. So this red curve or wave that is created is now my sine graph. So the cosine graph and the sine graphs are very similar. They’re just a translation of each other.

So this is where those graphs come from, if you can’t remember. You should always refer back to the unit circle. Now, what might be useful for us to look at next is maybe how those graphs might change. Maybe they can get bigger and smaller this way or bigger and smaller this way, depending on what values we put into the sine and the cosine function. So in our next video, we’ll look a little bit further into these functions, and also the tangent function. So, for now, let’s just summarise what we’ve done today. We’ve looked at how to measure angles in radians, and we’ll also be able to recognise the cosine and the sine graphs, given the unit circle.

Move on to our next video to explore these further.