0:00:01.000,0:00:03.600
Let's just do a ton of more examples, just so we[br]make sure that we're getting

0:00:03.600,0:00:07.000
this trig function thing down well.

0:00:07.000,0:00:10.733
So let's construct ourselves some right triangles.

0:00:10.733,0:00:15.533
Let's construct ourselves some right triangles, and I want to be very clear the way I've defined

0:00:15.533,0:00:18.867
it so far, this will only work in right triangles,[br]so if you're trying to find

0:00:18.867,0:00:24.400
the trig functions of angles that aren't part of right triangles, we're going to see that we're going to

0:00:24.400,0:00:27.533
have to construct right triangles, but let's just focus on the right triangles for now.

0:00:27.533,0:00:33.600
So let's say that I have a triangle, where[br]let's say this length down here is seven,

0:00:33.600,0:00:39.000
and let's say the length of this side up here, let's say that that is four.

0:00:39.000,0:00:43.333
Let's figure out what the hypotenuse over here is going to be. So we know

0:00:43.333,0:00:45.800
-let's call the hypotenuse "h"-

0:00:45.800,0:00:52.933
we know that h squared is going to be equal[br]to seven squared plus four squared, we know

0:00:52.933,0:00:55.533
that from of the Pythagorean theorem,

0:00:55.533,0:00:57.267
that the hypotenuse squared is equal to

0:00:57.267,0:01:00.333
the square of each of the sum of the squares

0:01:00.333,0:01:04.400
of the other two sides. Eight squared is equal to seven[br]squared plus four squared.

0:01:04.400,0:01:07.733
So this is equal to forty-nine

0:01:07.733,0:01:09.867
plus sixteen,

0:01:09.867,0:01:12.133
forty-nine plus sixteen,

0:01:12.133,0:01:16.267
forty nine plus ten is fifty-nine, plus[br]six is

0:01:16.267,0:01:21.667
sixty-five. It is sixty five so this h squared,

0:01:21.667,0:01:24.533
let me write: h squared

0:01:24.533,0:01:28.000
-that's different shade of yellow- so we have h squared is equal to

0:01:28.000,0:01:32.533
sixty-five. Did I do that right? Forty nine plus ten is fifty nine, plus another six

0:01:32.533,0:01:37.000
is sixty-five, or we could say that h is equal to, if we take the square root of

0:01:37.000,0:01:38.400
both sides

0:01:38.400,0:01:39.667
square root

0:01:39.667,0:01:43.067
square root of sixty five. And we really can't simplify[br]this at all

0:01:43.067,0:01:44.867
this is thirteen

0:01:44.867,0:01:49.267
this is the same thing as thirteen times five,[br]both of those are not perfect squares and

0:01:49.267,0:01:52.267
they're both prime so you can't simplify this any more.

0:01:52.267,0:01:54.733
So this is equal to the square root

0:01:54.733,0:01:56.400
of sixty five.

0:01:56.400,0:02:05.267
Now let's find the trig, let's find the trig functions for this angle[br]up here. Let's call that angle up there theta.

0:02:05.267,0:02:06.667
So whenever you do it

0:02:06.667,0:02:09.600
you always want to write down - at least for[br]me it works out to write down -

0:02:09.600,0:02:11.667
"soh cah toa".

0:02:11.667,0:02:13.333
soh...

0:02:13.333,0:02:15.533
...soh cah toa. I have these vague memories

0:02:15.533,0:02:18.000
of my

0:02:18.000,0:02:21.667
trigonometry teacher, maybe I've read it in some[br]book, I don't know - you know, some, about

0:02:21.667,0:02:25.200
some type of indian princess named "soh cah toa" or whatever, but it's a very useful

0:02:25.200,0:02:27.667
mnemonic, so we can apply "soh cah toa". Let's find

0:02:27.667,0:02:34.533
let's say we want to find the cosine. We want to find the cosine of our angle.

0:02:34.533,0:02:38.000
we wanna find the cosine of our angle, you[br]say: "soh cah toa!"

0:02:38.000,0:02:41.333
So the "cah". "Cah" tells us what to do with cosine,

0:02:41.333,0:02:43.400
the "cah" part tells us

0:02:43.400,0:02:46.533
that cosine is adjacent over hypotenuse.

0:02:46.533,0:02:49.933
Cosine is equal to adjacent

0:02:49.933,0:02:52.067
over hypotenuse.

0:02:52.067,0:02:56.000
So let's look over here to theta; what side is adjacent?

0:02:56.000,0:02:57.533
Well we know that the hypotenuse

0:02:57.533,0:03:00.867
we know that that hypotenuse is this side over here

0:03:00.867,0:03:05.133
so it can't be that side. The only other side that's kind of adjacent to it that

0:03:05.133,0:03:07.333
isn't the hypotenuse, is this four.

0:03:07.333,0:03:10.267
So the adjacent side over here, that side is,

0:03:10.267,0:03:14.333
it's literally right next to the angle, it's one of[br]the sides that kind of forms the angle

0:03:14.333,0:03:15.600
it's four

0:03:15.600,0:03:17.200
over the hypotenuse.

0:03:17.200,0:03:21.800
The hypotenuse we already know is square root[br]of sixty-five, so it's four

0:03:21.800,0:03:22.933
over

0:03:22.933,0:03:25.533
the square root of sixty-five.

0:03:25.533,0:03:29.933
And sometimes people will want you to rationalize[br]the denominator which means they don't like

0:03:29.933,0:03:34.267
to have an irrational number in the denominator,[br]like the square root of sixty five

0:03:34.267,0:03:36.867
and if they - if you wanna rewrite this without[br]a

0:03:36.867,0:03:41.667
irrational number in the denominator, you can[br]multiply the numerator and the denominator

0:03:41.667,0:03:43.333
by the square root of sixty-five.

0:03:43.333,0:03:47.400
This clearly will not change the number, because we're multiplying it by something over itself, so we're

0:03:47.400,0:03:51.533
multiplying the number by one. That won't change[br]the number, but at least it gets rid of the

0:03:51.533,0:03:54.000
irrational number in the denominator. So the numerator[br]becomes

0:03:54.000,0:03:58.067
four times the square root of sixty-five,

0:03:58.067,0:04:03.733
and the denominator, square root of sixty five times[br]square root of sixty-five, is just going to be sixty-five.

0:04:03.733,0:04:07.267
We didn't get rid of the irrational number, it's still[br]there, but it's now in the numerator.

0:04:07.267,0:04:09.400
Now let's do the other trig functions

0:04:09.400,0:04:13.800
or at least the other core trig functions. We'll[br]learn in the future that there's a ton of them

0:04:13.800,0:04:15.400
but they're all derived from these

0:04:15.400,0:04:20.200
so let's think about what the sign of theta is. Once again[br]go to "soh cah toa"

0:04:20.200,0:04:25.400
the "soh" tells what to do with sine. Sine is opposite over hypotenuse.

0:04:25.400,0:04:27.667
Sine is equal to

0:04:27.667,0:04:31.533
opposite over hypotenuse. Sine is opposite over hypotenuse.

0:04:31.533,0:04:34.600
So for this angle what side is opposite?

0:04:34.600,0:04:38.667
We just go opposite it, what it opens into, it's opposite[br]the seven

0:04:38.667,0:04:41.533
so the opposite side is the seven.

0:04:41.533,0:04:44.800
This right here - that is the opposite side

0:04:44.800,0:04:46.200
and then in the

0:04:46.200,0:04:49.267
hypotenuse, it's opposite over hypotenuse. the hypotenuse is the

0:04:49.267,0:04:50.867
square root of sixty-five

0:04:50.867,0:04:57.933
and once again if we wanted to rationalize this,[br]we could multiply times the square root of sixty-five

0:04:57.933,0:05:00.333
over the square root of sixty-five

0:05:00.333,0:05:06.067
and the the numerator, we'll get seven square root of sixty-five[br]and in the denominator we will get just

0:05:06.067,0:05:08.333
sixty-five again.

0:05:08.333,0:05:10.600
Now let's do tangent!

0:05:10.600,0:05:13.067
Let us do tangent.

0:05:13.067,0:05:14.933
So if i ask you the tangent

0:05:14.933,0:05:17.667
of - the tangent of theta

0:05:17.667,0:05:19.600
once again go back to soh cah

0:05:19.600,0:05:22.867
toa the toa part tells us what to do a tangent

0:05:22.867,0:05:25.000
it tells us

0:05:25.000,0:05:27.067
it tells us that tangent

0:05:27.067,0:05:31.000
is equal to opposite over adjacent is equal[br]to opposite

0:05:31.000,0:05:32.867
over

0:05:32.867,0:05:35.667
opposite over adjacent

0:05:35.667,0:05:37.267
so for this angle

0:05:37.267,0:05:41.800
what is opposite we've already figured it[br]out it's seven it opens into the seventh opposite

0:05:41.800,0:05:43.200
the seven

0:05:43.200,0:05:44.867
so it's seven

0:05:44.867,0:05:46.533
over what side is adjacent

0:05:46.533,0:05:48.267
well this four is adjacent

0:05:48.267,0:05:51.400
this four is adjacent so the adjacent side is[br]four

0:05:51.400,0:05:52.667
so it's seven

0:05:52.667,0:05:54.533
over four

0:05:54.533,0:05:55.800
and we're done

0:05:55.800,0:05:59.467
we figured out all of the trig ratios for[br]theta let's do another one

0:05:59.467,0:06:03.333
let's do another one. i'll make it a little bit concrete[br]'cause right now we've been saying oh was

0:06:03.333,0:06:06.600
tangent of x, tangent of theta. let's make it a little bit more concrete

0:06:06.600,0:06:08.800
let's say

0:06:08.800,0:06:11.400
let's say, let me draw another right triangle

0:06:11.400,0:06:14.333
that's another right triangle here

0:06:14.333,0:06:15.667
everything we're dealing with

0:06:15.667,0:06:18.333
these are going to be right triangles

0:06:18.333,0:06:19.667
let's say the hypotenuse

0:06:19.667,0:06:22.200
has length four

0:06:22.200,0:06:24.600
let's say that this side over here

0:06:24.600,0:06:26.933
has length two

0:06:26.933,0:06:32.067
and let's say that this length over here is goint to be two times the square root of three we can

0:06:32.067,0:06:33.800
verify that this works

0:06:33.800,0:06:38.267
if you have this side squared so you have let[br]me write it down two times the square root of

0:06:38.267,0:06:40.067
three squared

0:06:40.067,0:06:43.067
plus two squared is equal to what

0:06:43.067,0:06:44.333
this is

0:06:44.333,0:06:47.533
two there's going to be four times three

0:06:47.533,0:06:49.800
four times three plus four

0:06:49.800,0:06:55.067
and this is going to be equal to twelve plus[br]four is equal to sixteen and sixteen is indeed

0:06:55.067,0:06:58.267
four squared so this does equal four squared

0:06:58.267,0:07:02.000
it does equal four squared it satisfies the pythagorean theorem

0:07:02.000,0:07:06.933
and if you remember some of your work from thirty[br]sixty ninety triangles that you might have

0:07:06.933,0:07:09.133
learned in geometry you might recognize that[br]this

0:07:09.133,0:07:13.333
is a thirty sixty ninety triangle this[br]right here is our right angle i should have

0:07:13.333,0:07:16.400
drawn it from the get go to show that this[br]is a right triangle

0:07:16.400,0:07:20.400
this angle right over here is our thirty degree[br]angle

0:07:20.400,0:07:23.733
and then this angle up here, this angle up here[br]is

0:07:23.733,0:07:26.333
a sixty degree angle

0:07:26.333,0:07:28.267
and it's a thirty sixteen ninety because

0:07:28.267,0:07:32.200
the side opposite the thirty degrees is half the hypotenuse

0:07:32.200,0:07:37.200
and then the side opposite the sixty degrees[br]is a squared three times the other side

0:07:37.200,0:07:38.800
that's not the hypotenuse

0:07:38.800,0:07:42.267
so that's that we're not gonna this isn't supposed to be a review of thirty sixty ninety triangles

0:07:42.267,0:07:43.200
although i just did it

0:07:43.200,0:07:47.000
let's actually find the trig ratios[br]for the different angles

0:07:47.000,0:07:48.400
so if i were to ask you

0:07:48.400,0:07:50.667
or if anyone were to ask you what is

0:07:50.667,0:07:54.533
what is the sine of thirty degrees

0:07:54.533,0:07:58.933
and remember thirty degrees is one of the[br]angles in this triangle but it would apply

0:07:58.933,0:08:01.933
whenever you have a thirty degree angle and[br]you're dealing with the right triangle we'll

0:08:01.933,0:08:05.200
have broader definitions in the future but[br]if you say sine of thirty degrees

0:08:05.200,0:08:09.333
hey this ain't gold right over here is thirty[br]degrees so i can use this right triangle

0:08:09.333,0:08:12.267
and we just have to remember soh cah toa

0:08:12.267,0:08:14.400
rewrite it so

0:08:14.400,0:08:15.800
cah

0:08:15.800,0:08:17.333
toa

0:08:17.333,0:08:23.000
sine tells us soh tells us what to do with sine. sine is opposite over hypotenuse.

0:08:23.000,0:08:26.333
sine of thirty degrees is the opposite side

0:08:26.333,0:08:28.667
that is the opposite side which is two

0:08:28.667,0:08:32.267
over the hypotenuse. the hypotenuse here is four.

0:08:32.267,0:08:35.800
it is two fourths which is the same thing as[br]one-half

0:08:35.800,0:08:39.267
sine of thirty degrees you'll see is always going[br]to be equal

0:08:39.267,0:08:40.933
to one-half

0:08:40.933,0:08:42.400
now what is

0:08:42.400,0:08:44.267
the cosine

0:08:44.267,0:08:46.000
what is the cosine of

0:08:46.000,0:08:47.467
thirty degrees

0:08:47.467,0:08:50.333
once again go back to soh cah toa.

0:08:50.333,0:08:56.267
the cah tells us what to do with cosine. cosine is adjacent over hypotenuse

0:08:56.267,0:09:00.933
so for looking at the thirty degree angle[br]it's the adjacent this right over here is

0:09:00.933,0:09:02.133
adjacent it's right next to it

0:09:02.133,0:09:03.333
it's not the hypotenuse

0:09:03.333,0:09:06.467
it's the adjacent over the hypotenuse so[br]it's two

0:09:06.467,0:09:09.200
square roots of three

0:09:09.200,0:09:10.267
adjacent

0:09:10.267,0:09:11.333
over

0:09:11.333,0:09:13.733
over the hypotenuse over four

0:09:13.733,0:09:19.000
or if we simplify that we divide the numerator and the denominator by two it's the square root of three

0:09:19.000,0:09:20.667
over two

0:09:20.667,0:09:21.933
finally let's do

0:09:21.933,0:09:23.333
the tangent

0:09:23.333,0:09:27.733
tangent of thirty degrees

0:09:27.733,0:09:30.267
we go back to soh cah toa

0:09:30.267,0:09:31.600
soh cah toa

0:09:31.600,0:09:34.933
toa tells us what to do with tangent[br]it's opposite over adjacent

0:09:34.933,0:09:38.933
you go to the thirty degree angle because that's what we care about, tangent of thirty

0:09:38.933,0:09:43.000
tangent of thirty opposite is two

0:09:43.000,0:09:47.600
opposite is two and the adjacent is two square roots of three it's right next to it it's adjacent

0:09:47.600,0:09:48.333
to it

0:09:48.333,0:09:49.533
adjacent means next to

0:09:49.533,0:09:52.000
so two square roots of three

0:09:52.000,0:09:53.000
so this is equal to

0:09:53.000,0:09:56.600
the twos cancel out one over the square root[br]of three

0:09:56.600,0:10:00.200
or we could multiply the numerator and the denominator[br]by the square root of three

0:10:00.200,0:10:01.600
so we have

0:10:01.600,0:10:03.200
square root of three

0:10:03.200,0:10:05.333
over square root of three

0:10:05.333,0:10:09.733
and so this is going to be equal to the numerator[br]square root of three and then the denominator

0:10:09.733,0:10:14.667
right over here is just going to be three so[br]thats we've rationalized a square root of three

0:10:14.667,0:10:15.933
over three

0:10:15.933,0:10:17.000
fair enough

0:10:17.000,0:10:20.200
now lets use the same triangle to figure out the[br]trig ratios for the sixty degrees

0:10:20.200,0:10:22.200
since we've already drawn it

0:10:22.200,0:10:25.000
so what is

0:10:25.000,0:10:30.067
what is in the sine of the sixty degrees and i think you're hopefully getting the hang of it now

0:10:30.067,0:10:35.733
sine is opposite over adjacent. soh from the soh cah toa. from the sixty degree angle what side[br]

0:10:35.733,0:10:36.600
is opposite

0:10:36.600,0:10:42.600
what opens out into the two square roots of three[br]so the opposite side is two square roots of three

0:10:42.600,0:10:45.733
and from the sixty degree angle the adj-oh sorry its the

0:10:45.733,0:10:48.067
opposite over hypotenuse, don't want to confuse you.

0:10:48.067,0:10:50.533
so it is opposite over hypotenuse

0:10:50.533,0:10:54.600
so it's two square roots of three over four. four is the hypotenuse.

0:10:54.600,0:11:00.067
so it is equal to, this simplifies to square root of three over two.

0:11:00.067,0:11:05.600
what is the cosine of sixty degrees. cosine of sixty degrees.

0:11:05.600,0:11:10.133
so remember soh cah toa. cosine is adjacent over hypotenuse.

0:11:10.133,0:11:14.333
adjacent is the two sides right next to the sixty degree angle so it's two

0:11:14.333,0:11:17.800
over the hypotenuse which is four

0:11:17.800,0:11:19.600
so this is equal to

0:11:19.600,0:11:21.067
one-half

0:11:21.067,0:11:22.267
and then finally

0:11:22.267,0:11:26.333
what is the tangent, what is the tangent

0:11:26.333,0:11:28.000
of sixty degrees

0:11:28.000,0:11:32.267
well tangent soh cah toa tangent is opposite[br]over adjacent

0:11:32.267,0:11:34.600
opposite the sixty degrees

0:11:34.600,0:11:36.533
is two square roots of three

0:11:36.533,0:11:38.333
two square roots of three

0:11:38.333,0:11:40.000
and adjacent to that

0:11:40.000,0:11:41.267
adjacent to that

0:11:41.267,0:11:43.267
is two

0:11:43.267,0:11:45.200
adjacent to sixty degrees is two

0:11:45.200,0:11:46.933
so its opposite over adjacent

0:11:46.933,0:11:50.133
two square roots of three over two which is just equal to

0:11:50.133,0:11:52.867
the square root of three

0:11:52.867,0:11:54.800
And I just wanted to - look how these are related

0:11:54.800,0:12:01.364
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

0:12:01.364,0:12:05.848
and then these guys are the inverse of each other and i think if you think a little bit about this triangle

0:12:05.848,0:12:09.908
it will start to make sense why. we'll keep extending[br]this and give you a lot more practice in the next

0:12:09.908,9:59:59.000
few videos