0:00:00.859,0:00:04.478
Let's just do a ton of more examples, just so we[br]make sure that we're getting

0:00:04.478,0:00:07.020
this trig function thing down well.

0:00:07.020,0:00:11.109
So let's construct ourselves some right triangles.

0:00:11.109,0:00:15.299
Let's construct ourselves some right triangles, and I want to be very clear the way I've defined

0:00:15.299,0:00:19.829
it so far, this will only work in right triangles,[br]so if you're trying to find

0:00:19.829,0:00:24.204
the trig functions of angles that aren't part of right triangles, we're going to see that we're going to

0:00:24.204,0:00:28.079
have to construct right triangles, but let's just focus on the right triangles for now.

0:00:28.079,0:00:33.470
So let's say that I have a triangle, where[br]let's say this length down here is seven,

0:00:33.470,0:00:39.190
and let's say the length of this side up here, let's say that that is four.

0:00:39.190,0:00:43.170
Let's figure out what the hypotenuse over here is going to be. So we know

0:00:43.170,0:00:45.350
-let's call the hypotenuse "h"-

0:00:45.350,0:00:52.750
we know that h squared is going to be equal[br]to seven squared plus four squared, we know

0:00:52.750,0:00:55.110
that from of the Pythagorean theorem,

0:00:55.110,0:00:57.190
that the hypotenuse squared is equal to

0:00:57.190,0:01:00.289
the square of each of the sum of the squares

0:01:00.289,0:01:04.370
of the other two sides. Eight squared is equal to seven[br]squared plus four squared.

0:01:04.370,0:01:08.147
So this is equal to forty-nine

0:01:08.147,0:01:09.729
plus sixteen,

0:01:09.729,0:01:11.740
forty-nine plus sixteen,

0:01:11.740,0:01:16.851
forty nine plus ten is fifty-nine, plus[br]six is

0:01:16.851,0:01:21.979
sixty-five. It is sixty five so this h squared,

0:01:21.979,0:01:23.909
let me write: h squared

0:01:23.909,0:01:28.310
-that's different shade of yellow- so we have h squared is equal to

0:01:28.310,0:01:32.480
sixty-five. Did I do that right? Forty nine plus ten is fifty nine, plus another six

0:01:32.480,0:01:36.648
is sixty-five, or we could say that h is equal to, if we take the square root of

0:01:36.648,0:01:38.120
both sides

0:01:38.120,0:01:39.340
square root

0:01:39.340,0:01:42.850
square root of sixty five. And we really can't simplify[br]this at all

0:01:42.850,0:01:44.350
this is thirteen

0:01:44.350,0:01:48.819
this is the same thing as thirteen times five,[br]both of those are not perfect squares and

0:01:48.819,0:01:51.860
they're both prime so you can't simplify this any more.

0:01:51.860,0:01:54.673
So this is equal to the square root

0:01:54.673,0:01:56.248
of sixty five.

0:01:56.248,0:02:04.956
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.

0:02:05.060,0:02:06.270
So whenever you do it

0:02:06.270,0:02:09.679
you always want to write down - at least for[br]me it works out to write down -

0:02:09.679,0:02:11.219
"soh cah toa".

0:02:11.219,0:02:12.829
soh...

0:02:12.829,0:02:16.429
...soh cah toa. I have these vague memories

0:02:16.429,0:02:17.919
of my

0:02:17.919,0:02:21.029
trigonometry teacher, maybe I've read it in some[br]book, I don't know - you know, some, about

0:02:21.029,0:02:25.159
some type of indian princess named "soh cah toa" or whatever, but it's a very useful

0:02:25.159,0:02:28.079
pneumonic, so we can apply "soh cah toa". Let's find

0:02:28.079,0:02:34.099
let's say we want to find the cosine. We want to find the cosine of our angle.

0:02:34.099,0:02:37.779
we wanna find the cosine of our angle, you[br]say: "soh cah toa!"

0:02:37.779,0:02:41.219
So the "cah". "Cah" tells us what to do with cosine,

0:02:41.219,0:02:43.070
the "cah" part tells us

0:02:43.070,0:02:46.930
that cosine is adjacent over hypotenuse.

0:02:46.930,0:02:49.979
Cosine is equal to adjacent

0:02:49.979,0:02:51.529
over hypotenuse.

0:02:51.529,0:02:55.909
So let's look over here to theta; what side is adjacent?

0:02:55.909,0:02:57.759
Well we know that the hypotenuse

0:02:57.759,0:03:00.639
we know that that hypotenuse is this side over here

0:03:00.639,0:03:04.579
so it can't be that side. The only other side that's kind of adjacent to it that

0:03:04.579,0:03:06.949
isn't the hypotenuse, is this four.

0:03:06.949,0:03:10.479
So the adjacent side over here, that side is,

0:03:10.479,0:03:14.149
it's literally right next to the angle, it's one of[br]the sides that kind of forms the angle

0:03:14.149,0:03:15.269
it's four

0:03:15.269,0:03:16.669
over the hypotenuse.

0:03:16.669,0:03:21.929
The hypotenuse we already know is square root[br]of sixty-five, so it's four

0:03:21.929,0:03:22.470
over

0:03:22.470,0:03:25.130
the square root of sixty-five.

0:03:25.130,0:03:29.460
And sometimes people will want you to rationalize[br]the denominator which means they don't like

0:03:29.460,0:03:34.049
to have an irrational number in the denominator,[br]like the square root of sixty five

0:03:34.049,0:03:37.559
and if they - if you wanna rewrite this without[br]a

0:03:37.559,0:03:41.179
irrational number in the denominator, you can[br]multiply the numerator and the denominator

0:03:41.179,0:03:43.049
by the square root of sixty-five.

0:03:43.049,0:03:47.409
This clearly will not change the number, because we're multiplying it by something over itself, so we're

0:03:47.409,0:03:51.279
multiplying the number by one. That won't change[br]the number, but at least it gets rid of the

0:03:51.279,0:03:54.539
irrational number in the denominator. So the numerator[br]becomes

0:03:54.539,0:03:57.749
four times the square root of sixty-five,

0:03:57.749,0:04:03.280
and the denominator, square root of sixty five times[br]square root of sixty-five, is just going to be sixty-five.

0:04:03.280,0:04:07.129
We didn't get rid of the irrational number, it's still[br]there, but it's now in the numerator.

0:04:07.129,0:04:09.629
Now let's do the other trig functions

0:04:09.629,0:04:13.219
or at least the other core trig functions. We'll[br]learn in the future that there's a ton of them

0:04:13.219,0:04:15.249
but they're all derived from these

0:04:15.249,0:04:19.889
so let's think about what the sign of theta is. Once again[br]go to "soh cah toa"

0:04:19.889,0:04:25.650
the "soh" tells what to do with sine. Sine is opposite over hypotenuse.

0:04:25.650,0:04:27.383
Sine is equal to

0:04:27.383,0:04:31.509
opposite over hypotenuse. Sine is opposite over hypotenuse.

0:04:31.509,0:04:34.021
So for this angle what side is opposite?

0:04:34.021,0:04:38.930
We just go opposite it, what it opens into, it's opposite[br]the seven

0:04:38.930,0:04:41.909
so the opposite side is the seven.

0:04:41.909,0:04:44.349
This right here - that is the opposite side

0:04:44.349,0:04:45.710
and then in the

0:04:45.710,0:04:50.008
hypotenuse, it's opposite over hypotenuse. the hypotenuse is the

0:04:50.008,0:04:52.760
square root of sixty-five

0:04:52.760,0:04:57.989
and once again if we wanted to rationalize this,[br]we could multiply times the square root of sixty-five

0:04:57.989,0:05:00.469
over the square root of sixty-five

0:05:00.469,0:05:06.500
and the the numerator, we'll get seven square root of sixty-five[br]and in the denominator we will get just

0:05:06.500,0:05:08.089
sixty-five again.

0:05:08.089,0:05:10.219
Now let's do tangent!

0:05:10.219,0:05:12.479
Let us do tangent.

0:05:12.479,0:05:15.551
So if i ask you the tangent

0:05:15.551,0:05:17.330
of - the tangent of theta

0:05:17.330,0:05:19.979
once again go back to soh cah

0:05:19.979,0:05:23.120
toa the toa part tells us what to do a tangent

0:05:23.120,0:05:24.550
it tells us

0:05:24.550,0:05:27.319
it tells us that tangent

0:05:27.319,0:05:31.989
is equal to opposite over adjacent is equal[br]to opposite

0:05:31.989,0:05:33.379
over

0:05:33.379,0:05:35.639
opposite over adjacent

0:05:35.639,0:05:36.970
so for this angle

0:05:36.970,0:05:41.380
what is opposite we've already figured it[br]out it's seven it opens into the seventh opposite

0:05:41.380,0:05:42.549
the seven

0:05:42.549,0:05:44.409
so it's seven

0:05:44.409,0:05:46.089
over what side is adjacent

0:05:46.089,0:05:48.009
well this four is adjacent

0:05:48.009,0:05:51.040
this four is adjacent so the adjacent side is[br]four

0:05:51.040,0:05:52.639
so it's seven

0:05:52.639,0:05:54.049
over four

0:05:54.049,0:05:54.950
and we're done

0:05:54.950,0:05:59.349
we figured out all of the trig ratios for[br]theta let's do another one

0:05:59.349,0:06:03.129
let's do another one. i'll make it a little bit concrete[br]'cause right now we've been saying oh was

0:06:03.129,0:06:06.879
tangent of x, tangent of theta. let's make it a little bit more concrete

0:06:06.879,0:06:08.310
let's say

0:06:08.310,0:06:11.059
let's say, let me draw another right triangle

0:06:11.059,0:06:13.999
that's another right triangle here

0:06:13.999,0:06:15.250
everything we're dealing with

0:06:15.250,0:06:18.110
these are going to be right triangles

0:06:18.110,0:06:19.650
let's say the hypotenuse

0:06:19.650,0:06:21.919
has length four

0:06:21.919,0:06:24.440
let's say that this side over here

0:06:24.440,0:06:26.469
has length two

0:06:26.469,0:06:31.830
and let's say that this length over here is goint to be two times the square root of three we can

0:06:31.830,0:06:33.559
verify that this works

0:06:33.559,0:06:38.279
if you have this side squared so you have let[br]me write it down two times the square root of

0:06:38.279,0:06:40.039
three squared

0:06:40.039,0:06:42.930
plus two squared is equal to what

0:06:42.930,0:06:43.889
this is

0:06:43.889,0:06:47.119
two there's going to be four times three

0:06:47.119,0:06:49.549
four times three plus four

0:06:49.549,0:06:54.619
and this is going to be equal to twelve plus[br]four is equal to sixteen and sixteen is indeed

0:06:54.619,0:06:57.729
four squared so this does equal four squared

0:06:57.729,0:07:02.419
it does equal four squared it satisfies the pythagorean theorem

0:07:02.419,0:07:06.529
and if you remember some of your work from thirty[br]sixty ninety triangles that you might have

0:07:06.529,0:07:09.050
learned in geometry you might recognize that[br]this

0:07:09.050,0:07:13.030
is a thirty sixty ninety triangle this[br]right here is our right angle i should have

0:07:13.030,0:07:16.219
drawn it from the get go to show that this[br]is a right triangle

0:07:16.219,0:07:20.210
this angle right over here is our thirty degree[br]angle

0:07:20.210,0:07:24.430
and then this angle up here, this angle up here[br]is

0:07:24.430,0:07:26.019
a sixty degree angle

0:07:26.019,0:07:28.139
and it's a thirty sixteen ninety because

0:07:28.139,0:07:31.990
the side opposite the thirty degrees is half the hypotenuse

0:07:31.990,0:07:36.650
and then the side opposite the sixty degrees[br]is a squared three times the other side

0:07:36.650,0:07:38.280
that's not the hypotenuse

0:07:38.280,0:07:41.910
so that's that we're not gonna this isn't supposed to be a review of thirty sixty ninety triangles

0:07:41.910,0:07:43.110
although i just did it

0:07:43.110,0:07:46.830
let's actually find the trig ratios[br]for the different angles

0:07:46.830,0:07:48.080
so if i were to ask you

0:07:48.080,0:07:51.059
or if anyone were to ask you what is

0:07:51.059,0:07:54.389
what is the sine of thirty degrees

0:07:54.389,0:07:58.520
and remember thirty degrees is one of the[br]angles in this triangle but it would apply

0:07:58.520,0:08:01.520
whenever you have a thirty degree angle and[br]you're dealing with the right triangle we'll

0:08:01.520,0:08:04.970
have broader definitions in the future but[br]if you say sine of thirty degrees

0:08:04.970,0:08:10.099
hey this ain't gold right over here is thirty[br]degrees so i can use this right triangle

0:08:10.099,0:08:12.849
and we just have to remember soh cah toa

0:08:12.849,0:08:14.439
rewrite it so

0:08:14.439,0:08:15.949
cah

0:08:15.949,0:08:17.270
toa

0:08:17.270,0:08:22.159
sine tells us soh tells us what to do with sine. sine is opposite over hypotenuse.

0:08:23.050,0:08:26.199
sine of thirty degrees is the opposite side

0:08:26.199,0:08:29.279
that is the opposite side which is two

0:08:29.279,0:08:32.149
over the hypotenuse. the hypotenuse here is four.

0:08:32.149,0:08:35.800
it is two fourths which is the same thing as[br]one-half

0:08:35.800,0:08:39.020
sine of thirty degrees you'll see is always going[br]to be equal

0:08:39.020,0:08:40.760
to one-half

0:08:40.760,0:08:42.190
now what is

0:08:42.190,0:08:43.910
the cosine

0:08:43.910,0:08:45.980
what is the cosine of

0:08:45.980,0:08:47.160
thirty degrees

0:08:47.160,0:08:49.969
once again go back to soh cah toa.

0:08:49.969,0:08:56.070
the cah tells us what to do with cosine. cosine is adjacent over hypotenuse

0:08:56.070,0:08:59.940
so for looking at the thirty degree angle[br]it's the adjacent this right over here is

0:08:59.940,0:09:01.639
adjacent it's right next to it

0:09:01.639,0:09:02.960
it's not the hypotenuse

0:09:02.960,0:09:06.790
it's the adjacent over the hypotenuse so[br]it's two

0:09:06.790,0:09:08.779
square roots of three

0:09:08.779,0:09:10.320
adjacent

0:09:10.320,0:09:11.300
over

0:09:11.300,0:09:13.820
over the hypotenuse over four

0:09:13.820,0:09:19.290
or if we simplify that we divide the numerator and the denominator by two it's the square root of three

0:09:19.290,0:09:20.780
over two

0:09:20.780,0:09:23.200
finally let's do

0:09:23.200,0:09:25.880
the tangent

0:09:25.880,0:09:27.850
tangent of thirty degrees

0:09:27.850,0:09:29.179
we go back to soh cah toa

0:09:29.179,0:09:30.080
soh cah toa

0:09:30.080,0:09:34.900
toa tells us what to do with tangent[br]it's opposite over adjacent

0:09:34.900,0:09:38.860
you go to the thirty degree angle because that's what we care about, tangent of thirty

0:09:38.860,0:09:42.760
tangent of thirty opposite is two

0:09:42.760,0:09:47.150
opposite is two and the adjacent is two square roots of three it's right next to it it's adjacent

0:09:47.150,0:09:47.830
to it

0:09:47.830,0:09:49.430
adjacent means next to

0:09:49.430,0:09:51.720
so two square roots of three

0:09:51.720,0:09:53.110
so this is equal to

0:09:53.110,0:09:56.820
the twos cancel out one over the square root[br]of three

0:09:56.820,0:10:00.340
or we could multiply the numerator and the denominator[br]by the square root of three

0:10:00.340,0:10:01.740
so we have

0:10:01.740,0:10:03.290
square root of three

0:10:03.290,0:10:05.200
over square root of three

0:10:05.200,0:10:09.600
and so this is going to be equal to the numerator[br]square root of three and then the denominator

0:10:09.600,0:10:14.900
right over here is just going to be three so[br]thats we've rationalized a square root of three

0:10:14.900,0:10:15.890
over three

0:10:15.890,0:10:16.720
fair enough

0:10:16.720,0:10:20.500
now lets use the same triangle to figure out the[br]trig ratios for the sixty degrees

0:10:20.500,0:10:23.200
since we've already drawn it

0:10:23.200,0:10:24.890
so what is

0:10:24.890,0:10:30.580
what is in the sine of the sixty degrees and i think you're hopefully getting the hang of it now

0:10:30.580,0:10:35.480
sine is opposite over adjacent. soh from the soh cah toa. from the sixty degree angle what side[br]

0:10:35.480,0:10:36.760
is opposite

0:10:36.760,0:10:42.920
what opens out into the two square roots of three[br]so the opposite side is two square roots of three

0:10:42.920,0:10:47.880
and from the sixty degree angle the adj-oh sorry its the

0:10:47.880,0:10:54.420
opposite over hypotenuse, don't want to confuse you.

0:10:54.420,0:10:58.750
so it is opposite over hypotenuse

0:10:58.750,0:11:00.000
so it's two square roots of three over four. four is the hypotenuse.

0:11:00.000,0:11:03.139
so it is equal to, this simplifies to square root of three over two.

0:11:03.139,0:11:05.580
what is the cosine of sixty degrees. cosine of sixty degrees.

0:11:05.580,0:11:10.330
so remember soh cah toa. cosine is adjacent over hypotenuse.

0:11:10.330,0:11:15.070
adjacent is the two sides right next to the sixty degree angle so it's two

0:11:15.070,0:11:17.920
over the hypotenuse which is four

0:11:17.920,0:11:19.900
so this is equal to

0:11:19.900,0:11:20.860
one-half

0:11:20.860,0:11:22.120
and then finally

0:11:22.120,0:11:24.460
what is the tangent, what is the tangent

0:11:26.000,0:11:27.830
of sixty degrees

0:11:27.830,0:11:32.790
well tangent soh cah toa tangent is opposite[br]over adjacent

0:11:32.790,0:11:34.220
opposite the sixty degrees

0:11:34.220,0:11:36.130
is two square roots of three

0:11:36.130,0:11:37.940
two square roots of three

0:11:37.940,0:11:39.570
and adjacent to that

0:11:39.570,0:11:43.020
adjacent to that

0:11:43.020,0:11:45.470
is two

0:11:45.470,0:11:48.750
adjacent to sixty degrees is two

0:11:48.750,0:11:52.630
so its opposite over adjacent

0:11:52.630,0:11:56.000
two square roots of three over two which is just equal to

0:11:56.000,0:11:58.150
the square root of three

0:11:58.150,0:12:01.750
And I just wanted to - look how these are related

0:12:01.750,0:12:03.365
the sine of thirty degrees is the same as the cosine of sixty degrees

0:12:03.365,0:12:04.980
and then these guys are the inverse of each other and i think if you think a little bit about this triangle

0:12:05.440,0:12:09.519
it will start to make sense why. we'll keep extending[br]this and give you a lot more practice in the next

0:12:09.519,0:12:10.110
few videos