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[music] How do we really know that, that angle sum formula is true?

What happens if I just rotate a single point?

For instance, if I take this point here, the point (1, 0), and I rotate it through

an angle of theta, then I end up at the point (cosine theta, sine theta).

Somewhat similarly, if I take this point here, the point (0, 1), and I rotate

counterclockwise through an angle of theta, I end up at the point (minus sine

theta, cosine theta). What if I rotate some other point (x, 0)?

Well, if I rotate (x, 0), say x is some number between 0 and 1, well, then I just

think about where does (1, 0) go, right? (1,0) rotated at (cosine theta, sine

theta). So (x, 0) must rotate to (x cosine theta,

x sine theta), a point on the middle of this red line.

What if I rotate (0, y) through an angle of theta?

Similarly, if I've got some point, (0, y), say y is between 0 and 1, some point on

this red line, if i rotate that point through an angle theta, well the point (0,

1) rotates over to the point (minus sine theta, cosine theta) on the circle.

So the point (0, y) rotates to a point on the red line with coordinates (minus y

sine theta, y cosine theta). All right?

Its exactly this point, but scaled by y. Okay, so we know how to rotate (x, 0), and

we know how to rotate (0,y). How do i rotate (x, y)?

So, what if i rotate this point, (x, y), through an angle theta?

Well, lets rotate this, and all i really have to figure out is, where does the

point (x, 0) end up, and where does the point (0, y) end up?

Well, (x, 0) ends up at (x cosine theta, x sine theta).

This point started off as (x, 0), and when I rotate that through an angle theta, this

is where it ends up. And this point started off as (0, y), and

if I rotate that point through an angle of theta, this point ends up at (minus y sine

theta, y cosine theta). Now where does this point, (x, y) end up?

All I have to do is just add together these two coordinates to figure out the

coordinates of this other point. And that means that this point, (x, y)

ends up at x cosine theta minus y sine , and its y coordinate is x sine theta plus

y cosine theta. So now i know how to rotate a point.

How does that help? The goal was to try to justify the angle

sum formula. The trick is to think about what happens

when you do successive rotations. If you first rotate through and angle

alpha, and then an angle beta, that's the same as rotating through an angle alpha

plus beta all at once. Specifically, if I start with a point, (1,

0), and rotate through an angle of alpha, it ends up at (cosine alpha, sine alpha).

I could then take that point and rotate it to the angle beta, and I've got a formula

for that. Once I do that, the new coordinates are

cosine alpha cosine beta minus sine alpha sine beta, and then the y-coordinate is

cosine alpha sine beta plus sine alpha cosine beta.

But that's the same point as I'd get if I started with (1, 0), and just rotated the

whole thing initially through the angle alpha plus beta.

This means that cosine alpha plus beta must be equal to cosine alpha cosine beta

minus sine alpha sine beta, and sine alpha plus beta must be cosine alpha sine beta

plus sine alpha cosine beta. These angle sum formulas can look very

mysterious when they're written down, but they're really coming from a very simple

source. They're coming from the fact that rotating

through one angle and another angle is the same as rotating through the sum of those

two angles. .