- [Voiceover] I want to do a quick overview of trigonometry, and aspects of trig functions that are important to us as electrical engineers. So this isn't meant to be a full class on trigonometry. If you haven't had this subject before, this is something you can study on Khan Academy, and Sal does a lot of good videos on trig functions and how they work.
So the way I remember my trig functions, is with the phrase Soh cah toa. So this is the little phrase I use to remember how to do my trig functions. And we draw a triangle, like this, a right triangle, with an angle here of theta.
And we label the sides of the triangle as, this is the side adjacent to theta, this is the side, opposite of theta. And this side over here is the hypotenuse of the triangle. So this says that the definition of sine, of theta, equals opposite over hypotenuse.
Opposite over hypotenuse. This phrase here means the cosine, of theta, equals adjacent over hypotenuse. And the last one here is for the tangent.
It says the tangent of theta, equals opposite over adjacent. Opposite divided by adjacent. Opposite over adjacent.
So, soh cah toa helps you remember your trig functions. So lets take that idea over here, and draw a line out, and make some calculations. We have a graph here of a unit circle.
That means the radius of this is one everywhere. And what I want to know is, here's my angle theta. And angles are measured from the positive X axis.
Here's the X axis. Here's the Y axis. Angles are measured going counterclockwise.
So let's talk for a second about how angles are measured. Angles are measured in two ways. Angles are measured in degrees.
From zero to 360. And angles are also measured in something called radians. And that goes from zero to two pi.
So these are two different angle measures. And when you're measuring in degrees, we put the little degree mark up here. That's what that means.
Radians don't get a degree mark on them. So if I mark this out in degrees, here's zero degrees. Here's 90 degrees.
Here's 180 degrees. This is 270. And when I get back to the beginning, it's 360 degrees.
If I measure the same angles in radians, this'll be zero radians. When I get back here it's gonna be two pi radians. Going all the way around the circle is two pi radians.
That means that going half way around the circle is pi radians. That's equivalent to 180 degrees. If I do a quarter of a circle, that's equal to pi over two, radians.
And if I do three quarters of a circle, that's three halves pi, or three pi over two. So, we'll use degrees and radians all the time, and we'll flip back and forth between them. So, now let's do some trig functions, on our angle theta right in here.
Let's work out the sine, cosine, and tangent. Now let me give a name to this hypotenuse. Let's call that R.
And R equals one, right? I said this was a unit circle, so R is equal to one. And when we calculate our right triangle, what we do is we drop a perpendicular down here, to the X axis.
And we also draw a horizontal over here from the Y axis. This side right here, of the X axis right here is the side adjacent to the angle theta. And this distance right here on this side of our triangle, is the side opposite.
Okay, and basically there's gonna be a Y intercept here, and an X intercept right here, where those happen. These'll be some number, depending on the tilt of this line here. So, the sine, of theta, is equal to what?
Is equal to, let's look at our definition, it's equal to opposite over hypotenuse. Opposite is Y, over the hypotenuse which is R. If I look at cosine, theta, adjacent over hypotenuse.
Adjacent is the X distance, X distance, and the hypotenuse is R. And if we do the tangent, of theta, that equals what? It's opposite over adjacent.
So it's opposite, which is the Y distance, over X, the adjacent X, X is the adjacent, X. One thing to notice here about tangent, Y over X is the rise divided by the run, going from this point, up to this point. So that is the slope.
So the idea of slope and the idea of tangent, are really closely related. Just as one small point, let's work out, what is one radian? What's an angle of one radian in degrees?
I can do that conversion just doing some units. If we have 180 degrees, that equals pi, radians. So that means that one radian, equals 180 over pi.
If you plug that in the calculator, it'll come out to 57. 3, roughly, degrees. So, one radian actually is a little above 45 degrees.
One radian is 57 degrees. It looks about like that. We don't use this very often.
Mostly we talk about radians in terms of multiples of pi, 'cause it makes more sense on this circle. But, just to let you know, that's roughly one radian.
