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Let's just do a ton of more examples, just so we
make sure that we're getting

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this trig function thing down well.

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So let's construct ourselves some right triangles.

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Let's construct ourselves some right triangles, and I want to be very clear the way I've defined

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it so far, this will only work in right triangles,
so if you're trying to find

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the trig functions of angles that aren't part of right triangles, we're going to see that we're going to

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have to construct right triangles, but let's just focus on the right triangles for now.

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So let's say that I have a triangle, where
let's say this length down here is seven,

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and let's say the length of this side up here, let's say that that is four.

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Let's figure out what the hypotenuse over here is going to be. So we know

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-let's call the hypotenuse "h"-

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we know that h squared is going to be equal
to seven squared plus four squared, we know

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that from of the Pythagorean theorem,

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that the hypotenuse squared is equal to

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the square of each of the sum of the squares

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of the other two sides. h squared is equal to seven
squared plus four squared.

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So this is equal to forty-nine

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plus sixteen,

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forty-nine plus sixteen,

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forty nine plus ten is fifty-nine, plus
six is

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sixty-five. It is sixty five so this h squared,

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let me write: h squared

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-that's different shade of yellow- so we have h squared is equal to

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sixty-five. Did I do that right? Forty nine plus ten is fifty nine, plus another six

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is sixty-five, or we could say that h is equal to, if we take the square root of

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both sides

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square root

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square root of sixty five. And we really can't simplify
this at all

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this is thirteen

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this is the same thing as thirteen times five,
both of those are not perfect squares and

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they're both prime so you can't simplify this any more.

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So this is equal to the square root

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of sixty five.

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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.

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So whenever you do it

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you always want to write down - at least for
me it works out to write down -

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"soh cah toa".

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soh...

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...soh cah toa. I have these vague memories

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of my

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trigonometry teacher, maybe I've read it in some
book, I don't know - you know, some, about

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some type of indian princess named "soh cah toa" or whatever, but it's a very useful

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mnemonic, so we can apply "soh cah toa". Let's find

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let's say we want to find the cosine. We want to find the cosine of our angle.

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we wanna find the cosine of our angle, you
say: "soh cah toa!"

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So the "cah". "Cah" tells us what to do with cosine,

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the "cah" part tells us

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that cosine is adjacent over hypotenuse.

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Cosine is equal to adjacent

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over hypotenuse.

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So let's look over here to theta; what side is adjacent?

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Well we know that the hypotenuse

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we know that that hypotenuse is this side over here

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so it can't be that side. The only other side that's kind of adjacent to it that

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isn't the hypotenuse, is this four.

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So the adjacent side over here, that side is,

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it's literally right next to the angle, it's one of
the sides that kind of forms the angle

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it's four

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over the hypotenuse.

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The hypotenuse we already know is square root
of sixty-five, so it's four

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over

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the square root of sixty-five.

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And sometimes people will want you to rationalize
the denominator which means they don't like

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to have an irrational number in the denominator,
like the square root of sixty five

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and if they - if you wanna rewrite this without
a

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irrational number in the denominator, you can
multiply the numerator and the denominator

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by the square root of sixty-five.

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This clearly will not change the number, because we're multiplying it by something over itself, so we're

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multiplying the number by one. That won't change
the number, but at least it gets rid of the

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irrational number in the denominator. So the numerator
becomes

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four times the square root of sixty-five,

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and the denominator, square root of sixty five times
square root of sixty-five, is just going to be sixty-five.

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We didn't get rid of the irrational number, it's still
there, but it's now in the numerator.

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Now let's do the other trig functions

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or at least the other core trig functions. We'll
learn in the future that there's a ton of them

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but they're all derived from these

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so let's think about what the sign of theta is. Once again
go to "soh cah toa"

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the "soh" tells what to do with sine. Sine is opposite over hypotenuse.

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Sine is equal to

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opposite over hypotenuse. Sine is opposite over hypotenuse.

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So for this angle what side is opposite?

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We just go opposite it, what it opens into, it's opposite
the seven

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so the opposite side is the seven.

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This right here - that is the opposite side

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and then in the

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hypotenuse, it's opposite over hypotenuse. the hypotenuse is the

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square root of sixty-five

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and once again if we wanted to rationalize this,
we could multiply times the square root of sixty-five

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over the square root of sixty-five

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and the the numerator, we'll get seven square root of sixty-five
and in the denominator we will get just

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sixty-five again.

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Now let's do tangent!

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Let us do tangent.

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So if i ask you the tangent

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of - the tangent of theta

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once again go back to soh cah

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toa the toa part tells us what to do a tangent

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it tells us

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it tells us that tangent

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is equal to opposite over adjacent is equal
to opposite

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over

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opposite over adjacent

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so for this angle

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what is opposite we've already figured it
out it's seven it opens into the seventh opposite

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the seven

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so it's seven

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over what side is adjacent

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well this four is adjacent

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this four is adjacent so the adjacent side is
four

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so it's seven

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over four

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and we're done

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we figured out all of the trig ratios for
theta let's do another one

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let's do another one. i'll make it a little bit concrete
'cause right now we've been saying oh was

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tangent of x, tangent of theta. let's make it a little bit more concrete

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let's say

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let's say, let me draw another right triangle

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that's another right triangle here

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everything we're dealing with

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these are going to be right triangles

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let's say the hypotenuse

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has length four

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let's say that this side over here

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has length two

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and let's say that this length over here is goint to be two times the square root of three we can

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verify that this works

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if you have this side squared so you have let
me write it down two times the square root of

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three squared

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plus two squared is equal to what

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this is

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two there's going to be four times three

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four times three plus four

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and this is going to be equal to twelve plus
four is equal to sixteen and sixteen is indeed

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four squared so this does equal four squared

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it does equal four squared it satisfies the pythagorean theorem

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and if you remember some of your work from thirty
sixty ninety triangles that you might have

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learned in geometry you might recognize that
this

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is a thirty sixty ninety triangle this
right here is our right angle i should have

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drawn it from the get go to show that this
is a right triangle

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this angle right over here is our thirty degree
angle

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and then this angle up here, this angle up here
is

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a sixty degree angle

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and it's a thirty sixteen ninety because

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the side opposite the thirty degrees is half the hypotenuse

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and then the side opposite the sixty degrees
is a squared three times the other side

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that's not the hypotenuse

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so that's that we're not gonna this isn't supposed to be a review of thirty sixty ninety triangles

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although i just did it

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let's actually find the trig ratios
for the different angles

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so if i were to ask you

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or if anyone were to ask you what is

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what is the sine of thirty degrees

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and remember thirty degrees is one of the
angles in this triangle but it would apply

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whenever you have a thirty degree angle and
you're dealing with the right triangle we'll

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have broader definitions in the future but
if you say sine of thirty degrees

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hey this ain't gold right over here is thirty
degrees so i can use this right triangle

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and we just have to remember soh cah toa

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rewrite it so

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cah

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toa

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sine tells us soh tells us what to do with sine. sine is opposite over hypotenuse.

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sine of thirty degrees is the opposite side

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that is the opposite side which is two

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over the hypotenuse. the hypotenuse here is four.

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it is two fourths which is the same thing as
one-half

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sine of thirty degrees you'll see is always going
to be equal

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to one-half

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now what is

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the cosine

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what is the cosine of

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thirty degrees

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once again go back to soh cah toa.

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the cah tells us what to do with cosine. cosine is adjacent over hypotenuse

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so for looking at the thirty degree angle
it's the adjacent this right over here is

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adjacent it's right next to it

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it's not the hypotenuse

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it's the adjacent over the hypotenuse so
it's two

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square roots of three

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adjacent

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00:09:10,267 --> 00:09:11,333
over

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over the hypotenuse over four

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or if we simplify that we divide the numerator and the denominator by two it's the square root of three

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over two

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finally let's do

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the tangent

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tangent of thirty degrees

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we go back to soh cah toa

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soh cah toa

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toa tells us what to do with tangent
it's opposite over adjacent

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you go to the thirty degree angle because that's what we care about, tangent of thirty

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tangent of thirty opposite is two

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opposite is two and the adjacent is two square roots of three it's right next to it it's adjacent

193
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to it

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00:09:48,333 --> 00:09:49,533
adjacent means next to

195
00:09:49,533 --> 00:09:52,000
so two square roots of three

196
00:09:52,000 --> 00:09:53,000
so this is equal to

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00:09:53,000 --> 00:09:56,600
the twos cancel out one over the square root
of three

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00:09:56,600 --> 00:10:00,200
or we could multiply the numerator and the denominator
by the square root of three

199
00:10:00,200 --> 00:10:01,600
so we have

200
00:10:01,600 --> 00:10:03,200
square root of three

201
00:10:03,200 --> 00:10:05,333
over square root of three

202
00:10:05,333 --> 00:10:09,733
and so this is going to be equal to the numerator
square root of three and then the denominator

203
00:10:09,733 --> 00:10:14,667
right over here is just going to be three so
thats we've rationalized a square root of three

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00:10:14,667 --> 00:10:15,933
over three

205
00:10:15,933 --> 00:10:17,000
fair enough

206
00:10:17,000 --> 00:10:20,200
now lets use the same triangle to figure out the
trig ratios for the sixty degrees

207
00:10:20,200 --> 00:10:22,200
since we've already drawn it

208
00:10:22,200 --> 00:10:25,000
so what is

209
00:10:25,000 --> 00:10:30,067
what is in the sine of the sixty degrees and i think you're hopefully getting the hang of it now

210
00:10:30,067 --> 00:10:35,733
sine is opposite over adjacent. soh from the soh cah toa. from the sixty degree angle what side


211
00:10:35,733 --> 00:10:36,600
is opposite

212
00:10:36,600 --> 00:10:42,600
what opens out into the two square roots of three
so the opposite side is two square roots of three

213
00:10:42,600 --> 00:10:45,733
and from the sixty degree angle the adj-oh sorry its the

214
00:10:45,733 --> 00:10:48,067
opposite over hypotenuse, don't want to confuse you.

215
00:10:48,067 --> 00:10:50,533
so it is opposite over hypotenuse

216
00:10:50,533 --> 00:10:54,600
so it's two square roots of three over four. four is the hypotenuse.

217
00:10:54,600 --> 00:11:00,067
so it is equal to, this simplifies to square root of three over two.

218
00:11:00,067 --> 00:11:05,600
what is the cosine of sixty degrees. cosine of sixty degrees.

219
00:11:05,600 --> 00:11:10,133
so remember soh cah toa. cosine is adjacent over hypotenuse.

220
00:11:10,133 --> 00:11:14,333
adjacent is the two sides right next to the sixty degree angle so it's two

221
00:11:14,333 --> 00:11:17,800
over the hypotenuse which is four

222
00:11:17,800 --> 00:11:19,600
so this is equal to

223
00:11:19,600 --> 00:11:21,067
one-half

224
00:11:21,067 --> 00:11:22,267
and then finally

225
00:11:22,267 --> 00:11:26,333
what is the tangent, what is the tangent

226
00:11:26,333 --> 00:11:28,000
of sixty degrees

227
00:11:28,000 --> 00:11:32,267
well tangent soh cah toa tangent is opposite
over adjacent

228
00:11:32,267 --> 00:11:34,600
opposite the sixty degrees

229
00:11:34,600 --> 00:11:36,533
is two square roots of three

230
00:11:36,533 --> 00:11:38,333
two square roots of three

231
00:11:38,333 --> 00:11:40,000
and adjacent to that

232
00:11:40,000 --> 00:11:41,267
adjacent to that

233
00:11:41,267 --> 00:11:43,267
is two

234
00:11:43,267 --> 00:11:45,200
adjacent to sixty degrees is two

235
00:11:45,200 --> 00:11:46,933
so its opposite over adjacent

236
00:11:46,933 --> 00:11:50,133
two square roots of three over two which is just equal to

237
00:11:50,133 --> 00:11:52,867
the square root of three

238
00:11:52,867 --> 00:11:54,800
And I just wanted to - look how these are related

239
00:11:54,800 --> 00:12:01,364
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

240
00:12:01,364 --> 00:12:05,848
and then these guys are the inverse of each other and i think if you think a little bit about this triangle

241
00:12:05,848 --> 00:12:09,908
it will start to make sense why. we'll keep extending
this and give you a lot more practice in the next

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few videos