WEBVTT

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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. Eight 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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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

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to it

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adjacent means next to

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so two square roots of three

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so this is equal to

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the twos cancel out one over the square root
of three

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or we could multiply the numerator and the denominator
by the square root of three

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so we have

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square root of three

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over square root of three

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and so this is going to be equal to the numerator
square root of three and then the denominator

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right over here is just going to be three so
thats we've rationalized a square root of three

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over three

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fair enough

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now lets use the same triangle to figure out the
trig ratios for the sixty degrees

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since we've already drawn it

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so what is

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what is in the sine of the sixty degrees and i think you're hopefully getting the hang of it now

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sine is opposite over adjacent. soh from the soh cah toa. from the sixty degree angle what side


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is opposite

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what opens out into the two square roots of three
so the opposite side is two square roots of three

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and from the sixty degree angle the adj-oh sorry its the

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opposite over hypotenuse, don't want to confuse you.

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so it is opposite over hypotenuse

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so it's two square roots of three over four. four is the hypotenuse.

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so it is equal to, this simplifies to square root of three over two.

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what is the cosine of sixty degrees. cosine of sixty degrees.

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so remember soh cah toa. cosine is adjacent over hypotenuse.

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adjacent is the two sides right next to the sixty degree angle so it's two

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over the hypotenuse which is four

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so this is equal to

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one-half

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and then finally

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what is the tangent, what is the tangent

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of sixty degrees

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well tangent soh cah toa tangent is opposite
over adjacent

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opposite the sixty degrees

00:11:34.600 --> 00:11:36.533
is two square roots of three

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two square roots of three

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and adjacent to that

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adjacent to that

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is two

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adjacent to sixty degrees is two

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so its opposite over adjacent

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two square roots of three over two which is just equal to

00:11:50.133 --> 00:11:52.867
the square root of three

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And I just wanted to - look how these are related

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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

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

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

00:12:09.908 --> 99:59:59.999
few videos