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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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# Sine and cosine come from circles

Sine and cosine can be generated by projecting the tip of a vector onto the y-axis and x-axis as the vector rotates about the origin. Created by Willy McAllister.

## Want to join the conversation?

- Will drawing a circle with a bigger radius allow us to find angles that have sine or cosine values that are greater than 1?(1 vote)
- Clever idea, but no. Trig functions (sine, cosine, tangent) are defined as ratios. Sine is "opposite over hypotenuse" (the SOH of SOHCAHTOA). When we draw the triangle inside a unit circle the hypotenuse is automatically 1 at any angle. That means the sine of an angle is simply the length of the "opposite" leg of the triangle (opposite / 1). If you make the circle radius = 2 it makes both O and H twice as long, but the ratio stays the same. Nice try. I like the way you think.(4 votes)

- Hello, we are starting the curves of sine and cosine at school, is this what I need to know for an understanding of it, I mean is this the basics for starting the sine and cosine graphs/curves? Thank you(1 vote)
- Short answer: Yes.

When sine and cosine are first revealed in Trigonometry class they are taught as ratios of the the sides of right triangles. The memorization tool is "SOA CAH TOA".

If you fit the right triangle inside a circle, you can slide the tip of the triangle around the circle and think about sin and cos of a point somewhere along the circle. It all means the same thing, just a slightly different mental picture.

Using this circle perspective is really handy. In the next few videos we let the point on the circle move as time goes by, so you get this orbiting point. That turns out to be a super useful way to think about Signals. (sound, radio waves, light)

Meanwhile, in Trig class you will use the right triangle viewpoint to prove all sorts of useful properties and identities of trig functions.(3 votes)

- Can you please explain where did the sine function come from? From the unit circle? or exponentially?(0 votes)
- Why does the sine and the cosine waves look like waves and not like semicircles joining each other in succession if they are related to circles?(1 vote)
- Sine and cosine are
*related*to circles but are not*made*of circles. Recall the definition of sine is based on a right triangle inside a circle. sin = opposite / hypotenuse, and cos = adjacent / hypotenuse (SOH CAH TOA).(1 vote)

## Video transcript

- [Voiceover] Now I'm gonna
clear off the screen here and we're gonna talk about the
shape of the sine function. Let's do that. This is a plot of the sine function where the angle, theta, this
is the theta axis in this plot, where theta has been plotted
out on a straight line instead of wrapped around this circle. So if we draw a line on here, let's make this circle a radius one. So if I draw this line up here, it's on a unit circle,
the definition of sine of theta, this will be theta here, is opposite over hypotenuse. So this is the opposite side and that distance is the opposite leg of that triangle, is
this value right here. So sine of theta is actually equal to y over the hypotenuse and
the hypotenuse is one in all cases around this. So, if I plot this on a curve, this is an angle and I
basically go over here and plot it like that. And then as theta swings
around the circle, I'm gonna plot the different values of y. If it comes over this
way, down here like this, right, you can see that, that plots over there like that. Now when the angle gets
back all the way to zero, of course, the sine function
comes all the way back to zero and then it repeats again
as our vector swings around the other way. So the sine of two pi is zero,
just like the sine of zero. So every two pi, if I go off the screen, every two pi comes back
and repeats to zero. So now I wanna do the same
thing with the cosine function that we did with sine, where
we project the projection of this value onto this time
the cosine curve down here. This has the cosine curve with
time going down on the page. And our definition of cosine was adjacent over hypotenuse. Hypotenuse is one in our drawing. So cosine of theta equals adjacent which is x, the x value, divided by hypotenuse which is one. So in this diagram, the cosine of theta is actually the x value
which is this x right here. So let me clean this off for a second. And we'll start at the beginning. Let's start with the radius
pointing straight sideways. And we know that cosine of
theta equals zero is one. So if I drop that down,
if I project that down onto the angle zero, that's this point right here on the curve. Now as we roll forward,
we go to a higher angle, this projection now moves
to here on the curve. When the arrow is straight up, we are at this point right here, we go back to the axis. If we go continue on,
this projects down here. We're moving this radius vector around in a circle like this. And actually this one
will be at the same point as before, as the one above, but it'll be on this part of the curve here. And when we get back to zero again, the projection is to this point here. So that's a way to
visualize the cosine curve getting generated by a vector
rotating around this circle. The cosine comes out the bottom because it's the projection on the x-axis, and when we did the sine,
it was the projection on the y-axis, produced the
sine wave when we went this way. So I like to visualize this
because this rotating vector is a really simple and powerful idea, and we can see how it actually generates, it's a way to generate
sine and cosine waves. And you can see how sort of naturally they come out at different phases, right. The sine starts at zero and
the cosine starts at one. With this way of drawing it,
you could see why that happens. So this relationship between circles and rotating vectors and sines and cosines is a very powerful idea. We're really gonna take advantage of this.