Welcome back.
We'll now try to see what trigonometric identity we can
come up with if we start off with cosine of alpha plus beta.
Let's see if we can rewrite this as another combination
of cosines and sines of alpha and beta.
So let's get started.
And if you've already watched the sine equivalent of this,
this proof will be pretty similar in how we operate.
And we get a kind of similar answer.
And something interesting is to kind of compare the difference
between sine of alpha plus beta and cosine of alpha plus
beta after we're done.
So just like that last proof, let's say that this angle-- no,
that color isn't bright enough.
Let's do yellow.
Let's say that this angle right here is alpha, and this
angle right here is beta.
Right?
We want to know cosine of alpha plus beta.
So alpha plus beta is this large angle right here.
Right?
So what's the cosine of that?
SOH, CAH, TOA.
So cosine is adjacent over hypotenuse.
Right?
SOH, CAH, TOA.
CAH.
So cosine is equal to adjacent over hypotenuse.
So for this large angle, what's the adjacent?
It's line AC.
So that equals-- line AC, that's the adjacent.
The length of line AC over the length of line--
what's the hypotenuse?
AB, right?
All right.
AC over AB.
Now let's see what we can do with this.
AC-- adjacent over hypotenuse.
Now can we write AC in any other interesting way-- a
combination of some of the other lines on this very
fortunately designed graph?
Let's see.
Well isn't AC the same thing as AF, this big line,
minus-- what is this?
This is a D, right?
OK.
That's a D.
Let me rewrite that as a D.
D as in dog.
There you go.
So AF minus DE.
Right?
Oh, I forgot to draw some things.
We assume that this line is perpendicular to that line.
We assume this line is perpendicular to that line.
We assume that line is perpendicular to that line.
Right?
And then by definition, that is because we drew it that way.
But anyway.
So now you know that this line is parallel to this line and
this line is perpendicular.
So we know that AF, this long line, minus DE is equal to AC.
Does that make sense?
AF, this big line, minus the shorter line is
the same thing as AC.
Right?
So let me write that down.
That equals AF minus DE, all of that over AB.
And then, of course, we can rewrite that as-- and I'm going
to switch to some different colors-- as AF over AB.
Let me switch to maybe green.
Minus DE over AB.
So we have now AF over AB minus DE over AB.
And those are kind of nonsensical ratios to me.
Wouldn't it be great if we could express it
somehow as AF over AE?
Because then we could say well that's cosine of alpha, and
do something from there.
Well let's try.
So let's try to rewrite this first expression.
So I'll switch back to the purple just so you
know where this first expression is coming from.
Let's see if we can break this down as AF over AE
times something else.
Well, we could just algebraically do it.
That's equivalent to AF over-- I might run out of space--
over AE times AE over AB.
And you're saying, Sal where did you get that from?
Well, you can kind of say my motivation was to have AF
as a ratio of over AE.
And I just set it up so that the multiplication would cancel
out, because the AE's would cancel out and you'd be
left with AF over AB.
Right?
So this is a reasonable thing to do.
I hope you see.
And let me switch to the green and do something similar.
DE over AB doesn't make much sense.
But if I could maybe do DE over BE, then if this angle I can
see is similar to alpha or beta then maybe I can
make some progress.
So let's say DE over BE times-- and we'll do the same thing.
You just have to multiply times BE over AB.
And just like in that sine proof, and we'll do the same
thing here, let's figure out what this angle up here is.
Right?
Because if we know that then these ratios become useful.
So if this angle is an alpha, then we know that this
angle right here is alpha.
Right?
Because DE-- because this line-- is parallel to AF.
You learned that in geometry.
And if this angle is alpha we know that this angle right here
is 90 minus alpha, because it's complementary.
Right?
Because this whole angle is 90 degrees, so this
is 90 minus alpha.
And since this angle 90 minus alpha, this angle 90, and this
angle add up to 180, we could figure out that this is alpha.
And if you don't believe me add up alpha plus 90 plus 90 minus
alpha, and you will get 180 degrees.
So this angle up here, angle DBE, is alpha.
So that's very interesting.
So can we rewrite these ratios as the sines or cosines
of alpha's or beta's?
Well, let's try.
Let me switch back to purple.
So that equals-- what is AF over AE?
Well if we look at this right triangle, that's the adjacent
over the hypotenuse for alpha.
Right?
Adjacent over hypotenuse, that's cosine.
So it's cosine of alpha.
And what's AE over AB?
Well, they're similar.
If we look at this big right triangle right here that is the
adjacent over the hypotenuse for beta, so it's
cosine of beta.
Switch my colors.
Minus DE over BE.
Well this is alpha, right?
Now there's a little smudge.
You probably can't read it.
But that was alpha.
We showed that that was alpha.
So DE is the opposite and BE is the hypotenuse.
Opposite over hypotenuse is sine, right?
So that's sine of alpha.
And what is BE over AB?
Look at this triangle again.
Well, for beta that is BE is the opposite and
AB is the hypotenuse.
So opposite over hypotenuse for beta.
So it's the sine of beta.
Times the sine of beta.
I'm running out of space.
I have to go to another line.
Pretty neat.
I'll rewrite everything in a new and exciting color.
OK.
Let me do it in light blue.
So we now know that the cosine of alpha plus beta is equal to
the cosine of both of them multiplied-- so cosine of
alpha, cosine of beta-- minus the sine of both of
them multiplied.
Minus sine of alpha times sine of beta.
I hope you found that as satisfying as I do.
See you in the next presentation.