Welcome back.
I'm now going to do a bit of a review of everything we've
learned so far about maybe even trigonometry and
trig identities.
And then we'll see if we can come up with-- maybe use what
we already know to come up with a couple more trig identities.
So we know that from SOH-CAH-TOA we know that sine
of theta is equal to the opposite over the hypotenuse.
Let me draw a triangle here.
If I were to draw a triangle here-- whoops.
Oh, there you go.
OK, so this is theta.
This is the opposite.
This is the adjacent.
This is hypotenuse.
Then sine of theta is equal to opposite over hypotenuse.
Cosine of theta-- this is basic review, hopefully at this
point-- is the adjacent over the hypotenuse.
The tangent of theta is equal to the opposite over the
adjacent, which is also equal to the sine of theta over
the cosine of theta.
And we showed this in a couple of videos ago.
And then, these are kind of almost definitional, but the
cosecant of theta is equal to the hypotenuse over the
opposite, which is the same thing as 1 over sine of theta.
You can just memorize this.
I mean, I kind of find is silly that there is such
a thing as cosecant.
I guess it's just for convenience because you know
everyone knows it's just 1 over sine of theta.
And same thing for secant.
Secant of theta-- it's really for convenience.
Instead of having to say, in the case of secant-- oh, that's
1 you know-- if you end up with the equation 1 over cosine of
theta you can just say, oh, that's just the
secant of theta.
I think it actually has some obvious properties and if you
were to draw unit circle and all of that too.
But anyway, so that's equal to the hypotenuse over the
adjacent, which is equal to 1 over cosine of theta.
And then, of course, cotangent of theta is equal to the
adjacent over the opposite, which is equal to
1 over tan theta.
And of course, that's also equal to cosine of
theta over sine theta.
It's just the opposite of the tangent of theta.
Or that's the same thing as what?
That's the same thing as the secant-- no, no, no.
It's the same thing as the cosecant-- no, no no.
Let me make sure I get this right.
It's the same thing as the-- I just want to get the inverses.
Well, let's prove what it is actually.
I always confuse myself.
So this is the same thing as 1 over the secant of theta over
1 over the cosecant of theta.
Secant of theta, cosecant theta.
And then that equals the cosecant of theta over
the secant of theta.
I wouldn't waste your time memorizing.
So we know that a cotangent of theta is equal to
1 over tangent theta.
Is equal to the cosine over the sine.
And it also equals the cosecant over the secant.
And I wouldn't worry about really memorizing this.
You could derive it if you had to.
As you could tell, I really didn't have
this memorized either.
And we also learned in previous videos that the sine squared
theta plus the cosine squared of theta is equal to 1.
And that just comes from the pythagorean theorem.
And if you play around with this a little bit you'd also
get that the tangent squared theta plus 1 is equal to
the secant squared theta.
You actually go from here to here if you just divide both
sides of this equation by cosine squared.
So we know that.
And then if you've watched the last two proof videos I made,
we also know that the sine of-- let's say a plus b-- is equal
to the sine of a times the cosine of b.
Plus-- let me erase some of this because I don't
think that that is an important trig identity.
You can derive it on your own.
I just wanted to show you that you could figure it out.
I'm using too much space.
OK, now I have space.
Let me find that blue color I was using and make
sure my pen is small.
OK.
So it's the sine of a times the cosine of b plus the sine
of b times the cosine of a.
And you might want to just memorize it.
This actually becomes really useful when you actually start
doing calculus because you have to solve derivatives and
integrals that you might have to know the identity.
And it's not that hard to memorize.
It's the sine of one of them times the cosine of one of them
plus the other way around.
That's all this is.
And then we also learned that the cosine of a plus b-- it's
the cosine of both of them minus the sine of both them.
So that is equal to the cosine of a times the cosine of b.
And I proved this in another video, hopefully did it
to your satisfaction.
Minus the sine of a times the sine of b.
These are pretty useful because from these can we can come up
with a bunch of other trig identities.
For example, what is the sine of 2a?
Well, that's just the same thing as the sine of a plus a.
And if we use this trig identity up here, that is equal
to sine of a cosine of a plus the sine of a, cosine of a.
I just used this sine of a plus b identity up here and well,
want a and b are both a.
Now what does this equal?
Well, this is two terms that are just both sine
of a, cosine of a.
So that just equals 2 sine of a, cosine of a.
So we now have derived another trigonometric identity that
might be in the inside cover of your trig, or actually,
your calculus book.
All of these actually, I could draw a square
around all of them.
Let's do another one.
Once you have a bit of a library of trig identities you
can really just keep playing around and seeing what else you
can-- and I encourage you to do so.
And you'd be amazed how many other trig identities
you could come up with.
For example, let's do cosine of 2a.
Cosine of 2a is equal to cosine of a plus a.
And cosine of a plus a, what did we say?
It's the cosine of both of the terms times each other minus
the sine of both of the terms.
So that equals cosine of a, cosine of a, right?
Cosine of a times cosine of a minus sine of a, sine of a.
This identity was the cosine of a plus b identity.
Minus sine of a.
So what is this?
This is equal to cosine squared a minus sine squared a.
That's interesting.
We could play around.
This is interesting because this is the form a
squared minus b squared.
So that's also the same thing as a plus b times a minus b.
So that's the same thing as cosine of a plus sine of a
times cosine of a minus sine of a.
I don't know.
This isn't really a trig identity, but I'm just showing
you could play with things.
Cosine of 2a is equal to cosine of a plus sine of a times
cosine of a minus sine of a.
So the sum of the cosine and sine of a then
times the difference.
That's just interesting.
I'm just showing you that what's fun about trigonometry
is you can kind of keep playing around with it.
And actually, that's probably-- that is how all of the trig
identities were discovered.
So let's say that we have-- we want to figure out what cosine
of let's say, negative a is.
Well, let me draw a right triangle.
Whoops.
That's almost a right triangle.
Now let's say this angle is a.
So negative a, unit circle would look something
like this, right?
Negative a.
So cosine of a, if we say that this side is the adjacent
side, this is the hypotenuse.
This would still be the hypotenuse, right?
And this is the opposite.
This is the negative opposite.
So cosine of minus a is equal to what?
This is minus a, so it's the adjacent over the hypotenuse.
So it equals the adjacent over the hypotenuse,
which we just say is h.
But that's the same thing as cosine of a, right?
Because cosine of a is also the adjacent over the hypotenuse.
Oh, I'm almost out of time.
Let me switch to a new video.