Welcome back, because I was hitting against the
YouTube 10-minute limit.
But all I was saying is, we said, you know,
cosine of minus a.
So I drew a right triangle with a, and then I showed you minus
a, and I said, well, all of the lengths are going to be the
same, but now the direction of-- and we're kind of assuming
this is all on the unit circle.
If you don't remember the unit circle, maybe you'll want to
rewatch the videos we have on that.
But I'm just showing you that the cosine of minus a is equal
to this side over the hypotenuse, and this hypotenuse
is the same as this hypotenuse, right?
So cosine of minus a is adjacent over this hypotenuse,
while cosine of a is adjacent over this hypotenuse.
But it's the same thing, so we know that cosine of minus
a is equal to cosine of a.
Actually, by definition, that makes it a-- I don't want to
confuse you too much, but that makes cosine an even function,
and I'll show you more.
Actually, I should do a whole presentation on
even and odd functions.
Now, let's see what sine of minus a is.
Sine of minus a is equal to-- so this is minus a.
So it's this side, so it's the minus length of-- let's call
this x, let's call this y, and let's call this, well,
let's leave that h, right?
If that is x, this is y, this length is y, then this length
right here is minus x, right?
So the sine of minus a is minus x/h.
What's the sine of a?
Sine of a is equal to-- this is a-- opposite or hypotenuse,
x over h, right?
So sine of minus a is equal to minus 1 times x over h, right?
Or this is just the same thing as-- I mean, we could multiply
both sides of this by minus 1, minus x over h, right?
So sine of minus a is equal to minus sine of a.
So let me clear this out and rewrite this identity.
And as you can see, all I'm doing is I'm just playing
around with triangles and showing you that, you know,
just using the basic SOHCAHTOA, you can actually discover a
whole set of trigonometric identities.
So let's clear that.
And, you know, it is useful to memorize.
I normally don't advocate memorizing, but it's helpful
just to do things quickly.
But I'd also advocate being able to prove it to yourself,
so if you ever forget it, and you don't have a cheat sheet
available, you can prove it, and if you ever have to teach
it, then you'll be able to explain the underlying
themes a little bit better.
So let's clear this.
Let's see if we can discover some more trig identities.
So we know that-- so let's see, if we have sine--
what's sine of a plus pi/2?
a plus pi/2.
Well, we could use our handy sine of a plus b identity,
which we've already proved, so we can use it now.
So that tells us that it's the sine of a-- that equals the
sine of a times the cosine of pi/2 plus the sine of pi/2.
And we're in radians, of course.
This could have been 90 degrees instead, if we
wanted to be in degrees.
sine of pi/2 times the cosine of a, right?
Well, this equals the sine of -- what's cosine sign of pi/2?
Or cosine of 90 degrees?
Well, that's when we're on the unit circle, we're
pointing straight up.
And so the x-coordinate is 0.
I could draw it out, but I think-- you might want to draw
the unit circle and figure it out for yourself, or if you
don't, do it on a calculator, but you will learn
that it is 0.
The cosine of pi/2 is 0.
Plus sine of pi/2, for the same reason, we're pointing straight
up on the unit circle, so the y-coordinate, or the sine
coordinate, is 1, right on the unit-- is essentially at the
point 0, 1 on the unit circle.
So sine of pi/2 is 1, and then times cosine of a.
So sine of a times 0 is 0.
1 times cosine of a is just cosine of a.
So we have a new, useful trig identity.
Sine a plus pi/2 is equal to cosine of a.
Fascinating!
So really, this is just telling us that cosine of a is the same
thing as sine of a shifted.
So if we were to think of this graphically, if we were to
think of, you know, if we were to draw the graph, if you shift
the sine graph to the left by pi/2, you get the cosine graph.
And if you haven't learned about shifting yet,
don't worry about that.
Or you might want to actually graph the two, and I
think you'll get a sense of what I'm saying.
So let's do-- I don't know.
And another way to rewrite this exact same thing is the sine of
a is equal to the cosine of a minus pi/2, right?
Let's say I said that b is a plus pi/2, right?
Let's say I said that b is equal to a plus pi/2, then we
can say that this is b, and then this would b minus pi/2.
I'm just switching around variables.
I'm doing this in a much more loosey-goosey fashion than I
normally do a lot of videos, but I want to show you that a
lot of this trigonometry can just be-- you know, it's
just kind of discovery.
What's sine of a minus b?
Well, that looks like a new one, doesn't it?
Well, let's try to figure it out.
Well, that equals sine of a cosine of minus b plus sine
of minus b times the cosine of a, right?
Well, what do we know about the cosine of minus b?
Before I cleared the screen, we just figured out that the
cosine of minus b, since it's an even function, is the same
thing as the cosine of b.
So we can rewrite that as that equals the
sine of a cosine of b.
And then what's the sine of minus b?
Well, that's the same thing as the minus sine of b.
That's what we just proved, that the sine of minus
b, that this is equal to minus sine of b.
You could draw the triangle and the unit circle, if you don't
believe me, but we just did that.
So we can say that that is equal to minus sine
of b cosine of a.
Interesting!
I encourage you do the same thing with the
cosine of a minus b.
These are all just, you know, we're using one or two or three
trig identities together and trying to come up
with new things.
And I think at this point, we've literally gone over
everything, that almost every trig identity you've seen in
your book, you should be able to get there somehow,
just by keep on playing.
And obviously, all of these identities, you can invert the
sines and the cosines and the tangents, and you can get
identities for secant and cotangent and cosecant
and keep playing around.
And I encourage you to do so.
And do it graphically.
Draw the triangles.
It's also interesting to sometimes actually draw the
graph on the x-y plane of, say, you know, cosine of x plus
pi/2, or sine of x plus pi/2, or sine of x.
And I think in the future, I'll do a video where I really
do explore all of that.
Well, I hope I haven't thoroughly confused you.
I wanted to just kind of show you that a lot of trig-- it
all comes from SOHCAHTOA and playing around with SOHCAHTOA
and triangles, and you can pretty much get-- you can
pretty much solve for everything you learn
in trigonometry.
And if you don't have SOHCAHTOA, at least the unit
circle definition, which is actually better, because
it's more extensive.
But anyway, that's all for now.
See you soon.