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

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What happens if I just rotate a single
point?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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sine theta, y cosine theta).
All right?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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sum formula.
The trick is to think about what happens

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

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

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

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

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

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

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

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cosine alpha sine beta plus sine alpha
cosine beta.

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

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whole thing initially through the angle
alpha plus beta.

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

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

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

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

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source.
They're coming from the fact that rotating

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

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two angles.
.