Welcome back.
I'm now going to do a series of videos on the
trigonometric identities.
So let's just review what we already know about the
trig function, so let me just write SOHCAHTOA.
That tells us, and we've actually extended this with the
unit circle definition, but if you watch those videos, you'll
realize that the unit circle definition directly
uses SOHCAHTOA.
So we'll just stick with SOHCAHTOA because I think it'll
help make some of what we're about to do seem a little bit
more straightforward and will kind of verge on the unit
circle definition anyway.
So we know that sine of theta is equal to opposite
over hypotenuse, right?
So cosine of theta is equal to adjacent over hypotenuse, and
then the tangent of theta is equal to opposite
over adjacent.
So let's draw that out on a right triangle.
We could do this with the unit circle as well, and
it would make sense.
Let's see if we can find a relationship between sine,
cosine and tangent.
There's my right triangle.
Let's call this theta.
This is the hypotenuse h.
This is the opposite side, right, opposite of theta.
This is theta right here.
This is the adjacent side, right?
Well, what do we know about the relationship between
the opposite adjacent side and then the hypotenuse?
What does the Pythagorean theorem tell us?
Oh, yeah, this side squared plus this side squared is equal
to the hypotenuse squared, so we could write that down.
a squared plus o squared is equal to the hypotenuse
squared, right?
And then this is just an equation, so if we want to, we
could divide both sides of this equation by h squared,
and so what do we get?
We get a squared over h squared plus o squared over h squared
is equal to 1, right?
And then I could rewrite that as a over h squared plus o
over h squared is equal to 1.
Now, do these look at all familiar?
Well, we have them here, right?
This is a over h, this is o over h, so we could
just substitute.
So this is just cosine of theta squared.
And this is how you write cosine squared.
You could put a parentheses around the whole thing and then
square it, but this is just the notation people use.
Plus opposite over adjacent squared, so that's sine theta
squared is equal to 1.
So that's our first trig identity.
So if you know the sine of theta, it's very easy to figure
out the cosine of theta, right?
You could just solve this equation.
If I know that the-- I don't know.
Let's say I know that the sine of theta is 1/2, right?
Then what is the cosine of theta?
The cosine of theta is what?
Well, I know the sine of theta is 1/2, right?
So I would say cosine squared of theta plus sine of theta
is 1/2, so 1/2 squared is equal to 1, right?
So cosine squared theta plus 1/4 is equal to 1.
So we have cosine squared theta is equal to 3/4, or cosine of
theta would be the square root of this, right?
We just take the square root of both sides.
It would be the square root of 3/2.
And you probably remember that from our whole presentation
on the 30-60-90 triangle.
So I just wanted to show you a use of this trig identity
that's usually written sine squared plus cosine
squared is equal to 1.
So let's extend that one a little bit.
Let's just play with the ratios and see what else we can--
other identities we can discover.
Whoops!
Clear image, invert colors.
So we know that sine squared theta plus cosine squared
theta is equal to 1.
The one thing we could do is we could divide both sides of this
equation by cosine squared of theta, and let's just see what
happens when we do that.
So if we say cosine squared theta, right?
You have to distribute across both terms.
Cosine squared of theta, and then cosine squared of theta.
Well, what's sine squared theta over cosine squared theta?
That's the same thing as sine of theta over cosine of theta
squared plus this is 1 over cosine theta squared, right?
I mean, 1 squared is 1, so I just rewrote it.
So sine over cosine theta, I think we learned that already.
That's just the tangent of theta.
And in case you actually haven't learned that already,
think about it this way.
Sine is opposite over the hypotenuse, right?
So that's opposite over hypotenuse.
And then cosine is adjacent over hypotenuse.
So adjacent over hypotenuse.
So then that equals opposite over hypotenuse times
hypotenuse over adjacent, right?
Just dividing by a fraction is the same thing as multiplying
by its reciprocal.
That's all I did.
And that equals opposite over adjacent, right?
So that just says sine of theta over cosine of theta is
equal to tangent of theta.
So sine squared theta over cosine squared theta is tan
squared theta, then plus 1 is 1 over cosine theta squared.
And now I'm going to introduce a new trig ratio, it's really
just 1 over cosine theta.
So 1 over cosine theta-- and I'm going to summarize this at
the end, just so it's not too confusing-- is actually
the secant of theta.
And this is just another ratio, right?
The secant of theta, instead of being the adjacent over the
hypotenuse, would be the hypotenuse over the
adjacent, right?
It's just 1 over cosine theta.
Nothing fancy here.
So secant of theta.
So that equals secant squared of theta.
I know it can be a little overwhelming initially, just
because I'm, you know, throwing out all these new terms, secant
is 1 over cosine theta, but once you just play around with
these enough and get familiar with the terms, it'll make
sense, and it'll be a little more natural to you.
So this could be-- you could view this as
another trig identity.
And actually, I don't even remember if I've
taught it already.
I mean, you could view this as a trig identity, although
that's almost definitional.
And then, of course, you can-- in case I haven't done it
already, you now know that sine of theta over cosine of theta
is equal to tangent of theta.
And that's right here with, I guess you could
say, the proof of it.
So let me keep introducing you to more things, and if this is
really daunting, maybe you just can rewatch it, and
hopefully, it'll make sense.
Let me see, clear image.
So what have we learned so far?
We learned that sine squared theta plus cosine squared
theta is equal to 1.
We learned that sine of theta over cosine of theta is
equal to tangent of theta.
We learned that the tangent squared of theta plus 1 is
equal to the secant of theta.
And here, let me actually write this definition down.
The secant of theta-- oops, is equal to the secant
squared of theta, sorry.
And the secant of theta is just 1 over cosine of theta.
This is something you really should just memorize, that
secant is 1 over cosine.
And if you're wondering what 1 over sine is, 1 over sine of
theta, it's the cosecant-- the abbreviation is csc-- of theta.
And if you're wondering what 1 over the tangent is,
it's the cotangent.
And you just might want to memorize these.
And this often confuses me, that 1 over the cosine is the
secant, but 1 over the sine is the cosecant, so it's kind of
almost the opposite, right?
1 over the sine has a co in it, while 1 over the cosine
doesn't have the co in it.
So that might help you remember things.
So I think that's all I have time for now.
In the next presentation, I'm going to introduce you to a
couple more trig identities.
See you soon.