Let's just do a ton of more examples, just so we
make sure that we're getting
this trig function thing down well.
So let's construct ourselves some right triangles.
Let's construct ourselves some right triangles, and I want to be very clear.
The way I've defined it so far, this will only work in right triangles.
So if you're trying to find the trig functions of angles that aren't part of right triangles,
we're going to see that we're going to have to construct right triangles,
but let's just focus on the right triangles for now.
So let's say that I have a triangle, where
let's say this length down here is seven,
and let's say the length of this side up here, let's say that that is four.
Let's figure out what the hypotenuse over here is going to be.
So we know -let's call the hypotenuse, "h"-
we know that h squared is going to be equal
to seven squared plus four squared,
we know that from the Pythagorean theorem,
that the hypotenuse squared is equal to
the square of each of the sum of the squares of the other two sides.
h squared is equal to seven
squared plus four squared.
So this is equal to forty-nine plus sixteen,
forty-nine plus sixteen,
forty nine plus ten is fifty-nine, plus
six is sixty-five.
It is sixty five. So this h squared,
let me write: h squared -that's different shade of yellow-
so we have h squared is equal to sixty-five.
Did I do that right? Forty nine plus ten is fifty nine, plus another six is sixty-five,
or we could say that h is equal to, if we take the square root of both sides,
square root
square root of sixty five. And we really can't simplify
this at all.
This is thirteen.
This is the same thing as thirteen times five,
both of those are not perfect squares and
they're both prime so you can't simplify this any more.
So this is equal to the square root of sixty five.
Now let's find the trig, let's find the trig functions for this angle up here.
Let's call that angle up there theta.
So whenever you do it
you always want to write down - at least for
me it works out to write down -
"soh cah toa".
soh...
...soh cah toa. I have these vague memories
of my trigonometry teacher.
Maybe I've read it in some book. I don't know - you know, some... about
some type of indian princess named "soh cah toa" or whatever,
but it's a very useful mnemonic,
so we can apply "soh cah toa".
Let's find, let's say we want to find the cosine.
We want to find the cosine of our angle.
We wanna find the cosine of our angle, you
say: "soh cah toa!"
So the "cah". "Cah" tells us what to do with cosine,
the "cah" part tells us
that cosine is adjacent over hypotenuse.
Cosine is equal to adjacent over hypotenuse.
So let's look over here to theta; what side is adjacent?
Well we know that the hypotenuse,
we know that that hypotenuse is this side over here.
So it can't be that side. The only other side that's kind of adjacent to it that
isn't the hypotenuse, is this four.
So the adjacent side over here, that side is,
it's literally right next to the angle, it's one of
the sides that kind of forms the angle
it's four over the hypotenuse.
The hypotenuse we already know is square root
of sixty-five.
so it's four over the square root of sixty-five.
And sometimes people will want you to rationalize
the denominator which means
they don't like to have an irrational number in the denominator,
like the square root of sixty five,
and if they - if you wanna rewrite this without
a irrational number in the denominator,
you can
multiply the numerator and the denominator
by the square root of sixty-five.
This clearly will not change the number, because we're multiplying it by something over itself,
so we're multiplying the number by one.
That won't change the number, but at least it gets rid of the irrational number in the denominator.
So the numerator
becomes
four times the square root of sixty-five,
and the denominator, square root of sixty five times
square root of sixty-five, is just going to be sixty-five.
We didn't get rid of the irrational number, it's still
there, but it's now in the numerator.
Now let's do the other trig functions
or at least the other core trig functions. We'll
learn in the future that there's actually a ton of them
but they're all derived from these.
so let's think about what the sign of theta is. Once again
go to "soh cah toa".
The "soh" tells what to do with sine. Sine is opposite over hypotenuse.
Sine is equal to opposite over hypotenuse.
Sine is opposite over hypotenuse.
So for this angle what side is opposite?
We just go opposite it, what it opens into, it's opposite
the seven
so the opposite side is the seven.
This is, right here - that is the opposite side
and then the hypotenuse, it's opposite over hypotenuse.
The hypotenuse is the square root of sixty-five.
Square root of sixty-five.
and once again if we wanted to rationalize this,
we could multiply times the square root of sixty-five over the square root of sixty-five
and the the numerator, we will get seven square root of sixty-five
and in the denominator we will get just sixty-five again.
Now let's do tangent!
Let us do tangent.
So if i ask you the tangent
of - the tangent of theta
once again go back to "soh cah toa".
The toa part tells us what to do with tangent
it tells us...
it tells us that tangent
is equal to opposite over adjacent
is equal to opposite over
opposite over adjacent
So for this angle, what is opposite? We've already figured it
out.
it's seven. It opens into the seven.
It is opposite the seven.
So it's seven over what side is adjacent.
well this four is adjacent.
This four is adjacent. So the adjacent side is
four.
so it's seven over four,
and we're done.
We figured out all of the trig ratios for
theta. let's do another one.
Let's do another one. i'll make it a little bit concrete
'cause right now we've been saying,
"oh, what's tangent of x, tangent of theta." let's make it a little bit more concrete.
Let's say...
let's say, let me draw another right triangle,
that's another right triangle here.
Everything we're dealing with, these are going to be right triangles.
let's say the hypotenuse has length four,
let's say that this side over here has length two,
and let's say that this length over here is going to be two times the square root of three.
We can verify that this works.
If you have this side squared, so you have - let
me write it down -
two times the square root of three squared
plus two squared, is equal to what?
this is two. There's going to be four times three.
four times three plus four,
and this is going to be equal to twelve plus
four is equal to sixteen
and sixteen is indeed four squared. So this does equal four squared,
it does equal four squared. It satisfies the pythagorean theorem
and if you remember some of your work from thirty
sixty ninety triangles
that you might have learned in geometry,
you might recognize that this is a thirty sixty ninety triangle.
This
right here is our right angle,
- i should have drawn it from the get go to show that this
is a right triangle -
this angle right over here is our thirty degree
angle
and then this angle up here, this angle up here
is
a sixty degree angle,
and it's a thirty sixteen ninety because
the side opposite the thirty degrees is half the hypotenuse
and then the side opposite the sixty degrees
is a squared of three times the other side
that's not the hypotenuse.
So that said, we're not gonna this isn't supposed to be a review of thirty sixty ninety triangles although i just did it.
Let's actually find the trig ratios for the different angles.
So if i were to ask you or if anyone were to ask you, what is...
what is the sine of thirty degrees?
and remember thirty degrees is one of the
angles in this triangle but it would apply
whenever you have a thirty degree angle and
you're dealing with the right triangle.
We'll have broader definitions in the future but
if you say sine of thirty degrees,
hey, this angle right over here is thirty
degrees so i can use this right triangle,
and we just have to remember "soh cah toa"
We rewrite it. soh, cah, toa.
"sine tells us" (correction). soh tells us what to do with sine. sine is opposite over hypotenuse.
sine of thirty degrees is the opposite side,
that is the opposite side which is two over the hypotenuse.
The hypotenuse here is four.
it is two fourths which is the same thing as
one-half.
sine of thirty degrees you'll see is always going
to be equal to one-half.
now what is the cosine?
What is the cosine of thirty degrees?
Once again go back to "soh cah toa".
The cah tells us what to do with cosine.
Cosine is adjacent over hypotenuse.
So for looking at the thirty degree angle
it's the adjacent.
This, right over here is adjacent. it's right next to it.
it's not the hypotenuse. it's the adjacent over the hypotenuse.
so it's two square roots of three
adjacent over...over the hypotenuse, over four.
or if we simplify that, we divide the numerator and the denominator by two
it's the square root of three over two.
Finally, let's do the tangent.
The tangent of thirty degrees,
we go back to "soh cah toa".
soh cah toa
toa tells us what to do with tangent. It's opposite over adjacent
you go to the thirty degree angle because that's what we care about, tangent of thirty.
tangent of thirty. Opposite is two,
opposite is two and the adjacent is two square roots of three.
It's right next to it it's adjacent to it.
adjacent means next to.
so two square roots of three
so this is equal to... the twos cancel out
one over the square root
of three
or we could multiply the numerator and the denominator
by the square root of three.
So we have square root of three over square root of three
and so this is going to be equal to the numerator
square root of three and then
the denominator right over here is just going to be three.
So that we've rationalized a square root of three over three.
Fair enough.
Now lets use the same triangle to figure out the
trig ratios for the sixty degrees,
since we've already drawn it.
so what is... what is the sine of the sixty degrees?
and i think you're hopefully getting the hang of it now.
Sine is opposite over adjacent. soh from the "soh cah toa".
for the sixty degree angle what side is opposite?
what opens out into the two square roots of three,
so the opposite side is two square roots of three,
and from the sixty degree angle the adj-oh sorry
its the opposite over hypotenuse, don't want to confuse you.
so it is opposite over hypotenuse
so it's two square roots of three over four. four is the hypotenuse.
so it is equal to, this simplifies to square root of three over two.
What is the cosine of sixty degrees? cosine of sixty degrees.
so remember "soh cah toa". cosine is adjacent over hypotenuse.
adjacent is the two sides, right next to the sixty degree angle.
So it's two over the hypotenuse which is four.
So this is equal to one-half
and then finally, what is the tangent?
what is the tangent of sixty degrees?
Well tangent, "soh cah toa". Tangent is opposite
over adjacent
opposite the sixty degrees
is two square roots of three
two square roots of three
and adjacent to that
adjacent to that is two.
Adjacent to sixty degrees is two.
So its opposite over adjacent, two square roots of three over two
which is just equal to the square root of three.
And I just wanted to -look how these are related-
the sine of thirty degrees is the same as the cosine of sixty degrees.
The cosine of thirty degrees is the same thing as the sine of sixty degrees
and then these guys are the inverse of each other and i think if you think a little bit about this triangle
it will start to make sense why.
we'll keep extending
this and give you a lot more practice in the next few videos.