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### Course: Electrical engineering > Unit 2

Lesson 5: AC circuit analysis- AC analysis intro 1
- AC analysis intro 2
- Trigonometry review
- Sine and cosine come from circles
- Sine of time
- Sine and cosine from rotating vector
- Lead Lag
- Complex numbers
- Multiplying by j is rotation
- Complex rotation
- Euler's formula
- Complex exponential magnitude
- Complex exponentials spin
- Euler's sine wave
- Euler's cosine wave
- Negative frequency
- AC analysis superposition
- Impedance
- Impedance vs frequency
- ELI the ICE man
- Impedance of simple networks
- KVL in the frequency domain

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# Trigonometry review

A quick review of some ideas from trigonometry that will help us with AC analysis. The definitions of sine, cosine, and tangent, circles, degrees, radians. Created by Willy McAllister.

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## Video transcript

- [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.