Trigonometry review

I want to do a quick overview of trigonometry and the 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 that you can study on KH 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 SOA TOA. This is the little phrase I use to remember how to do my trig functions. We draw a triangle like this, a right triangle with an angle here of theta. We label the sides of the triangle as follows: 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 that cosine of theta equals adjacent over hypotenuse. And the last one here is for the tangent; it says that the tangent of theta equals opposite over adjacent, opposite divided by adjacent, opposite over adjacent.

So SOA TOA helps you remember your trig functions. Let's take that idea over here and draw a line out and make some calculations. We have a graph here of our unit circle. That means the radius of this is one everywhere. 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 and 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 0 to 360, and angles are also measured in something called radians, and that goes from 0 to 2 pi.

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 0°, here's 90°, here's 180°, this is 270°, and when I get back to the beginning, it's 360°.

If I measure the same angles in radians, this will be zero radians. Well, when I get back here, it's going to be 2 pi radians. Going all the way around the circle is 2 pi radians. That means going halfway around the circle is pi radians; that's equivalent to 180°. If I do a quarter of a circle, that's equal to pi/2 radians, and if I do 3/4 of a circle, that's 3 pi/2 radians.

So we'll use degrees and radians all the time, and we'll flip back and forth between them. 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. When we calculate our right triangle, what we do is 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, this section of the x-axis, is the side adjacent to the angle Theta. This side, this distance right here on this side of our triangle, is the side opposite. Okay, and basically there's going to be a y-intercept here and a little x-intercept right here where those happen. These will be some number depending on the tilt of this angle of this line here.

So the sine of theta is equal to what? 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, and the hypotenuse is R.

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 is the adjacent x. Now, 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. 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 by 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 roughly 57.3 degrees. So one radian actually is a little above 45 degrees. One radian is 57°, and it looks about like that.

We don't use this very often; mostly we talk about radians in terms of multiples of pi because it makes more sense on this circle. But just to let you know, that's roughly one radian.