Hello! Well, welcome to I guess the next presentation in the trigonometry modules. Just to start of a little bit let us review what we have done so far. In the last couple of modules, we learned the definitions or at least I guess we call it a partial definition of the sine, the cosine, and tangent functions and mnemonic we used to memorize that was soh-cah-toa. And what that told us is let’s say we had a right triangle, this is the hypotenuse let me just write, let me label the hypotenuse H, let me label this and let us put this. Actually, we want to figure out. We want to use this angle right here, theta we will call this theta or whatever then this is the adjacent side and this is the opposite side and that is an O. And so it tells us that sine is equal to opposite over hypotenuse, cosine is equal to adjacent over hypotenuse and tangent is equal to opposite over adjacent.
Now, I think by this point especially if you did some of the exercises in the Khan Academy, that should be second nature and should make a lot of sense to you but this definition using a right triangle like this actually breaks down at certain points and actually a lot of points. For example what happens as this angle right here approaches 90 degrees and you cannot have two 90-degree angles in a right triangle can you? Then it would be like a rectangle or something but you can actually figure out what happens as it approaches 90 degrees but this definition breaks down for that. Also, what happens if this angle is negative or what happens if this angle is more than 90 degrees or what happens if it is 800 degrees or you know 8π radians, not that 800 and 8π radians are the same thing but obviously this definition starts to break down because we could not even draw a right triangle that has this properties.
So now, I am going to introduce you to an extension of this definition. It is really the same thing but it allows the sine, the cosine, and the tangent functions to be defined for angles greater than or equal to π/2 or 90 degrees or less than 0. So, let us draw unit circle so this is just the coordinate axis and here is the circle of radius one. And let us make, let me see let me make sure I am using the correct pen tool, okay. So let us call this right here, so this is theta, this is an angle between this line, between kind of the x-axis and this line I just drew here and this is a radius right? And we said that this has a radius one so the length of this line is one right? Because it just goes from origin to the outside of the circle so it has a radius of one.
And now, I am going to draw a right triangle again so let me just draw up a line from here, so there I have a right triangle. So, if we use the old definition we learned before, let us just focus on sine for now. So, sine is equal to opposite over hypotenuse. Let us apply that to this right triangle right here, this is the right angle so what is the opposite angle of this? What is the opposite side from this angle? Right, I am going to change to yellow, it is this side right, this is the opposite side. And what is the hypotenuse? The hypotenuse is just this radius right? And let us just say that this point where it intersects the circle, let us call this point right here, let us call this point (x,y) alright.
So this, what is the height of this opposite side? Well it is y right because it is just the height of that point. This is a height y so sine of this angle right here, sine of theta is going to equal the opposite side which is this yellow side which is just the y-coordinate. It is going to equal y over the hypotenuse. The hypotenuse is this pink side here and what is the length of the hypotenuse? What’s the radius of this unit circle? So, it is one. And y divided by one, well that is just y. So, we see that sine of theta, sine of theta is equal to y. Let us do the same thing for cosine of theta. Well, we know that cosine is equal to adjacent over hypotenuse. Well, what is the adjacent side here? The adjacent side is this bottom side right here so that would equal—so if I said cosine of theta would equal to this gray side which is the adjacent side and what is that? What is this length? What is the length of this side? Well, it is just x right? If this is the point (x,y) then this distance here is x and we already learned this distance or we already observed that this distance is y. So, this distance being just x, we know that the adjacent length is x so we say cosine theta is equal to x over the hypotenuse and once again the hypotenuse is one. So, cosine of theta is equal to x.
I know what you are thinking, that is very nice and cute cosine of theta equals x, sine of theta equals y but how is this really different from what we were doing before? Well, if I define it this way, now all of a sudden when the angle becomes let us say 90 degrees, let us say this is, now I can actually define what sine of theta is. Sine of theta now is just y, it is just the y-coordinate which is 1. If theta is equal to, I am going to make sure it’s very messy right here. If theta is equal to 90 degrees or π/2 radians, right this is π/2, this angle right here, right? And similar cosine of π/2 is zero because the x-coordinate right here is zero. Let me do it with a couple more examples, I am forgetting the tangent function and you could probably figure out now what is the definition now we could use for the tangent function.
Well, going back, let us use this green theta here because it is kind of a normal angle so in this green theta here, tangent is opposite over adjacent. So tangent now, we can define as y over x. And remember these y’s and x’s that we are using are the point on the unit circle where the angle that is defined by whatever that where the radius that is subtended by this angle or I guess the arc intersects actually I am going to confuse the terminology, it is where this line intersects the circumference, the coordinate of that, the sine of theta is equal to y, the cosine of theta is equal to x and the tangent of theta is equal to y over x.
Let us do a couple of examples and hopefully this will make a little bit more sense to you. Let me try to really fast draw a new unit circle. So, that is my unit circle and here is the coordinate axes, that is one of them and here is the other one. So, let us get an example like—so if we use the angle, let us use the angle π/2 right? Theta equals π/2, well π/2 is right here, right? It is a 90-degree angle if you wanted to use degrees. And now we just figured out where it intersects the unit circle and once again this is a unit circle so it has radius of one. So, we can see that sine of π/2, it equals the y-coordinate where it intersects the unit circle so that is just one. What is cosine of π/2? Well, it is just the x-coordinate where you intersect the unit circle and the x-coordinate here is zero. And what is the tangent of π/2? This is interesting, the tangent of π/2, well the tangent we defined now as y/x so the y-coordinate, this is the point (0,1) right? The y coordinate is one so it equals one over zero? So, this is undefined. So still, we do not have a tangent function that can define itself at certain points but in the next module, we are actually going to graph this and you will see that it approaches infinity.
And similarly, we could try to find the functions for when theta equals π, that is like a 180 degrees. That is this point right here so sine of π, what is the y-coordinate at this point? Well, this point is (-1,0) so the y-coordinate is zero. What is the x-coordinate? Cosine of π, that is negative one. And of course what is the tangent of π radians? It is y/x, so it is zero over -1 which equals zero. Hopefully this makes sense. Now, in the next module I will actually graph these points and you will see how it all comes together and why it is useful to define the sine, the cosine, and tangent functions this way. See you soon, bye!
Transcription by:
Scribe4you Transcription Services