-
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 will only work in right triangles,
so if you try to find
-
the trig
-
functions of angles that aren't part of right
triangles, we're goint to see a 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 of the Pythagorean theorem.
-
That the hypotenuse squared is equal to
-
the square of each of the - the sum of the
squares
-
of the other two sides. Eight squared is equal to seven
squared plus four squared.
-
So this is equal to forty-nine
-
plus sixteen,
-
fourty-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 right: 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 see 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 is 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 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 about
-
some type of indian princess named "soh cah toa" or whatever, but it's a very useful
-
pneumonic, so we can apply "soh cah toa" to 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" 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, it's 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
the
-
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 denominator square two sixty five times
square to 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 will
learn in the future that there's a ton of them
-
but they're all derived from these
-
so let's think about 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 in the
-
hypotenuse, it's opposite over hypotenuse. the hypotenuse is the
-
square root sixty-five
-
and once again if we want to rationalize this
we could multiply times the square root of sixty- five
-
over the square root of sixty-five
-
and the numerator will get seven square roots 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
-
the tangent of theta
-
once again go back to soh cah
-
toa the toa part tells us what to do a 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 seventh 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 was
-
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 goint 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 three times the other side
-
that's not the hypotenuse
-
so that's that 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 ain't gold right over here is thirty
degrees so i can use this right triangle
-
and we just have to remember soh cah toa
-
rewrite it so
-
cah
-
toa
-
sine tells us 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
-
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
thats 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 in 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. from 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
-
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