[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. .