WEBVTT

00:00:00.859 --> 00:00:04.478
Let's just do a ton of more examples, just so we
make sure that we're getting

00:00:04.478 --> 00:00:07.020
this trig function thing down well.

00:00:07.020 --> 00:00:11.109
So let's construct ourselves some right triangles.

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

00:00:15.299 --> 00:00:19.829
it so far, this will only work in right triangles,
so if you're trying to find

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

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

00:00:28.079 --> 00:00:33.470
So let's say that I have a triangle, where
let's say this length down here is seven,

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

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

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

00:00:45.350 --> 00:00:52.750
we know that h squared is going to be equal
to seven squared plus four squared, we know

00:00:52.750 --> 00:00:55.110
that from of the Pythagorean theorem,

00:00:55.110 --> 00:00:57.190
that the hypotenuse squared is equal to

00:00:57.190 --> 00:01:00.289
the square of each of the sum of the squares

00:01:00.289 --> 00:01:04.370
of the other two sides. Eight squared is equal to seven
squared plus four squared.

00:01:04.370 --> 00:01:08.147
So this is equal to forty-nine

00:01:08.147 --> 00:01:09.729
plus sixteen,

00:01:09.729 --> 00:01:11.740
forty-nine plus sixteen,

00:01:11.740 --> 00:01:16.851
forty nine plus ten is fifty-nine, plus
six is

00:01:16.851 --> 00:01:21.979
sixty-five. It is sixty five so this h squared,

00:01:21.979 --> 00:01:23.909
let me write: h squared

00:01:23.909 --> 00:01:28.310
-that's different shade of yellow- so we have h squared is equal to

00:01:28.310 --> 00:01:32.480
sixty-five. Did I do that right? Forty nine plus ten is fifty nine, plus another six

00:01:32.480 --> 00:01:36.648
is sixty-five, or we could say that h is equal to, if we take the square root of

00:01:36.648 --> 00:01:38.120
both sides

00:01:38.120 --> 00:01:39.340
square root

00:01:39.340 --> 00:01:42.850
square root of sixty five. And we really can't simplify
this at all

00:01:42.850 --> 00:01:44.350
this is thirteen

00:01:44.350 --> 00:01:48.819
this is the same thing as thirteen times five,
both of those are not perfect squares and

00:01:48.819 --> 00:01:51.860
they're both prime so you can't simplify this any more.

00:01:51.860 --> 00:01:54.673
So this is equal to the square root

00:01:54.673 --> 00:01:56.248
of sixty five.

00:01:56.248 --> 00:02:04.956
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.

00:02:05.060 --> 00:02:06.270
So whenever you do it

00:02:06.270 --> 00:02:09.679
you always want to write down - at least for
me it works out to write down -

00:02:09.679 --> 00:02:11.219
"soh cah toa".

00:02:11.219 --> 00:02:12.829
soh...

00:02:12.829 --> 00:02:16.429
...soh cah toa. I have these vague memories

00:02:16.429 --> 00:02:17.919
of my

00:02:17.919 --> 00:02:21.029
trigonometry teacher, maybe I've read it in some
book, I don't know - you know, some, about

00:02:21.029 --> 00:02:25.159
some type of indian princess named "soh cah toa" or whatever, but it's a very useful

00:02:25.159 --> 00:02:28.079
pneumonic, so we can apply "soh cah toa". Let's find

00:02:28.079 --> 00:02:34.099
let's say we want to find the cosine. We want to find the cosine of our angle.

00:02:34.099 --> 00:02:37.779
we wanna find the cosine of our angle, you
say: "soh cah toa!"

00:02:37.779 --> 00:02:41.219
So the "cah". "Cah" tells us what to do with cosine,

00:02:41.219 --> 00:02:43.070
the "cah" part tells us

00:02:43.070 --> 00:02:46.930
that cosine is adjacent over hypotenuse.

00:02:46.930 --> 00:02:49.979
Cosine is equal to adjacent

00:02:49.979 --> 00:02:51.529
over hypotenuse.

00:02:51.529 --> 00:02:55.909
So let's look over here to theta; what side is adjacent?

00:02:55.909 --> 00:02:57.759
Well we know that the hypotenuse

00:02:57.759 --> 00:03:00.639
we know that that hypotenuse is this side over here

00:03:00.639 --> 00:03:04.579
so it can't be that side. The only other side that's kind of adjacent to it that

00:03:04.579 --> 00:03:06.949
isn't the hypotenuse, is this four.

00:03:06.949 --> 00:03:10.479
So the adjacent side over here, that side is,

00:03:10.479 --> 00:03:14.149
it's literally right next to the angle, it's one of
the sides that kind of forms the angle

00:03:14.149 --> 00:03:15.269
it's four

00:03:15.269 --> 00:03:16.669
over the hypotenuse.

00:03:16.669 --> 00:03:21.929
The hypotenuse we already know is square root
of sixty-five, so it's four

00:03:21.929 --> 00:03:22.470
over

00:03:22.470 --> 00:03:25.130
the square root of sixty-five.

00:03:25.130 --> 00:03:29.460
And sometimes people will want you to rationalize
the denominator which means they don't like

00:03:29.460 --> 00:03:34.049
to have an irrational number in the denominator,
like the square root of sixty five

00:03:34.049 --> 00:03:37.559
and if they - if you wanna rewrite this without
a

00:03:37.559 --> 00:03:41.179
irrational number in the denominator, you can
multiply the numerator and the denominator

00:03:41.179 --> 00:03:43.049
by the square root of sixty-five.

00:03:43.049 --> 00:03:47.409
This clearly will not change the number, because we're multiplying it by something over itself, so we're

00:03:47.409 --> 00:03:51.279
multiplying the number by one. That won't change
the number, but at least it gets rid of the

00:03:51.279 --> 00:03:54.539
irrational number in the denominator. So the numerator
becomes

00:03:54.539 --> 00:03:57.749
four times the square root of sixty-five,

00:03:57.749 --> 00:04:03.280
and the denominator, square root of sixty five times
square root of sixty-five, is just going to be sixty-five.

00:04:03.280 --> 00:04:07.129
We didn't get rid of the irrational number, it's still
there, but it's now in the numerator.

00:04:07.129 --> 00:04:09.629
Now let's do the other trig functions

00:04:09.629 --> 00:04:13.219
or at least the other core trig functions. We'll
learn in the future that there's a ton of them

00:04:13.219 --> 00:04:15.249
but they're all derived from these

00:04:15.249 --> 00:04:19.889
so let's think about what the sign of theta is. Once again
go to "soh cah toa"

00:04:19.889 --> 00:04:25.650
the "soh" tells what to do with sine. Sine is opposite over hypotenuse.

00:04:25.650 --> 00:04:27.383
Sine is equal to

00:04:27.383 --> 00:04:31.509
opposite over hypotenuse. Sine is opposite over hypotenuse.

00:04:31.509 --> 00:04:34.021
So for this angle what side is opposite?

00:04:34.021 --> 00:04:38.930
We just go opposite it, what it opens into, it's opposite
the seven

00:04:38.930 --> 00:04:41.909
so the opposite side is the seven.

00:04:41.909 --> 00:04:44.349
This right here - that is the opposite side

00:04:44.349 --> 00:04:45.710
and then in the

00:04:45.710 --> 00:04:50.008
hypotenuse, it's opposite over hypotenuse. the hypotenuse is the

00:04:50.008 --> 00:04:52.760
square root of sixty-five

00:04:52.760 --> 00:04:57.989
and once again if we wanted to rationalize this,
we could multiply times the square root of sixty-five

00:04:57.989 --> 00:05:00.469
over the square root of sixty-five

00:05:00.469 --> 00:05:06.500
and the the numerator, we'll get seven square root of sixty-five
and in the denominator we will get just

00:05:06.500 --> 00:05:08.089
sixty-five again.

00:05:08.089 --> 00:05:10.219
Now let's do tangent!

00:05:10.219 --> 00:05:12.479
Let us do tangent.

00:05:12.479 --> 00:05:15.551
So if i ask you the tangent

00:05:15.551 --> 00:05:17.330
of - the tangent of theta

00:05:17.330 --> 00:05:19.979
once again go back to soh cah

00:05:19.979 --> 00:05:23.120
toa the toa part tells us what to do a tangent

00:05:23.120 --> 00:05:24.550
it tells us

00:05:24.550 --> 00:05:27.319
it tells us that tangent

00:05:27.319 --> 00:05:31.989
is equal to opposite over adjacent is equal
to opposite

00:05:31.989 --> 00:05:33.379
over

00:05:33.379 --> 00:05:35.639
opposite over adjacent

00:05:35.639 --> 00:05:36.970
so for this angle

00:05:36.970 --> 00:05:41.380
what is opposite we've already figured it
out it's seven it opens into the seventh opposite

00:05:41.380 --> 00:05:42.549
the seven

00:05:42.549 --> 00:05:44.409
so it's seven

00:05:44.409 --> 00:05:46.089
over what side is adjacent

00:05:46.089 --> 00:05:48.009
well this four is adjacent

00:05:48.009 --> 00:05:51.040
this four is adjacent so the adjacent side is
four

00:05:51.040 --> 00:05:52.639
so it's seven

00:05:52.639 --> 00:05:54.049
over four

00:05:54.049 --> 00:05:54.950
and we're done

00:05:54.950 --> 00:05:59.349
we figured out all of the trig ratios for
theta let's do another one

00:05:59.349 --> 00:06:03.129
let's do another one. i'll make it a little bit concrete
'cause right now we've been saying oh was

00:06:03.129 --> 00:06:06.879
tangent of x, tangent of theta. let's make it a little bit more concrete

00:06:06.879 --> 00:06:08.310
let's say

00:06:08.310 --> 00:06:11.059
let's say, let me draw another right triangle

00:06:11.059 --> 00:06:13.999
that's another right triangle here

00:06:13.999 --> 00:06:15.250
everything we're dealing with

00:06:15.250 --> 00:06:18.110
these are going to be right triangles

00:06:18.110 --> 00:06:19.650
let's say the hypotenuse

00:06:19.650 --> 00:06:21.919
has length four

00:06:21.919 --> 00:06:24.440
let's say that this side over here

00:06:24.440 --> 00:06:26.469
has length two

00:06:26.469 --> 00:06:31.830
and let's say that this length over here is goint to be two times the square root of three we can

00:06:31.830 --> 00:06:33.559
verify that this works

00:06:33.559 --> 00:06:38.279
if you have this side squared so you have let
me write it down two times the square root of

00:06:38.279 --> 00:06:40.039
three squared

00:06:40.039 --> 00:06:42.930
plus two squared is equal to what

00:06:42.930 --> 00:06:43.889
this is

00:06:43.889 --> 00:06:47.119
two there's going to be four times three

00:06:47.119 --> 00:06:49.549
four times three plus four

00:06:49.549 --> 00:06:54.619
and this is going to be equal to twelve plus
four is equal to sixteen and sixteen is indeed

00:06:54.619 --> 00:06:57.729
four squared so this does equal four squared

00:06:57.729 --> 00:07:02.419
it does equal four squared it satisfies the pythagorean theorem

00:07:02.419 --> 00:07:06.529
and if you remember some of your work from thirty
sixty ninety triangles that you might have

00:07:06.529 --> 00:07:09.050
learned in geometry you might recognize that
this

00:07:09.050 --> 00:07:13.030
is a thirty sixty ninety triangle this
right here is our right angle i should have

00:07:13.030 --> 00:07:16.219
drawn it from the get go to show that this
is a right triangle

00:07:16.219 --> 00:07:20.210
this angle right over here is our thirty degree
angle

00:07:20.210 --> 00:07:24.430
and then this angle up here, this angle up here
is

00:07:24.430 --> 00:07:26.019
a sixty degree angle

00:07:26.019 --> 00:07:28.139
and it's a thirty sixteen ninety because

00:07:28.139 --> 00:07:31.990
the side opposite the thirty degrees is half the hypotenuse

00:07:31.990 --> 00:07:36.650
and then the side opposite the sixty degrees
is a squared three times the other side

00:07:36.650 --> 00:07:38.280
that's not the hypotenuse

00:07:38.280 --> 00:07:41.910
so that's that we're not gonna this isn't supposed to be a review of thirty sixty ninety triangles

00:07:41.910 --> 00:07:43.110
although i just did it

00:07:43.110 --> 00:07:46.830
let's actually find the trig ratios
for the different angles

00:07:46.830 --> 00:07:48.080
so if i were to ask you

00:07:48.080 --> 00:07:51.059
or if anyone were to ask you what is

00:07:51.059 --> 00:07:54.389
what is the sine of thirty degrees

00:07:54.389 --> 00:07:58.520
and remember thirty degrees is one of the
angles in this triangle but it would apply

00:07:58.520 --> 00:08:01.520
whenever you have a thirty degree angle and
you're dealing with the right triangle we'll

00:08:01.520 --> 00:08:04.970
have broader definitions in the future but
if you say sine of thirty degrees

00:08:04.970 --> 00:08:10.099
hey this ain't gold right over here is thirty
degrees so i can use this right triangle

00:08:10.099 --> 00:08:12.849
and we just have to remember soh cah toa

00:08:12.849 --> 00:08:14.439
rewrite it so

00:08:14.439 --> 00:08:15.949
cah

00:08:15.949 --> 00:08:17.270
toa

00:08:17.270 --> 00:08:22.159
sine tells us soh tells us what to do with sine. sine is opposite over hypotenuse.

00:08:23.050 --> 00:08:26.199
sine of thirty degrees is the opposite side

00:08:26.199 --> 00:08:29.279
that is the opposite side which is two

00:08:29.279 --> 00:08:32.149
over the hypotenuse. the hypotenuse here is four.

00:08:32.149 --> 00:08:35.800
it is two fourths which is the same thing as
one-half

00:08:35.800 --> 00:08:39.020
sine of thirty degrees you'll see is always going
to be equal

00:08:39.020 --> 00:08:40.760
to one-half

00:08:40.760 --> 00:08:42.190
now what is

00:08:42.190 --> 00:08:43.910
the cosine

00:08:43.910 --> 00:08:45.980
what is the cosine of

00:08:45.980 --> 00:08:47.160
thirty degrees

00:08:47.160 --> 00:08:49.969
once again go back to soh cah toa.

00:08:49.969 --> 00:08:56.070
the cah tells us what to do with cosine. cosine is adjacent over hypotenuse

00:08:56.070 --> 00:08:59.940
so for looking at the thirty degree angle
it's the adjacent this right over here is

00:08:59.940 --> 00:09:01.639
adjacent it's right next to it

00:09:01.639 --> 00:09:02.960
it's not the hypotenuse

00:09:02.960 --> 00:09:06.790
it's the adjacent over the hypotenuse so
it's two

00:09:06.790 --> 00:09:08.779
square roots of three

00:09:08.779 --> 00:09:10.320
adjacent

00:09:10.320 --> 00:09:11.300
over

00:09:11.300 --> 00:09:13.820
over the hypotenuse over four

00:09:13.820 --> 00:09:19.290
or if we simplify that we divide the numerator and the denominator by two it's the square root of three

00:09:19.290 --> 00:09:20.780
over two

00:09:20.780 --> 00:09:23.200
finally let's do

00:09:23.200 --> 00:09:25.880
the tangent

00:09:25.880 --> 00:09:27.850
tangent of thirty degrees

00:09:27.850 --> 00:09:29.179
we go back to soh cah toa

00:09:29.179 --> 00:09:30.080
soh cah toa

00:09:30.080 --> 00:09:34.900
toa tells us what to do with tangent
it's opposite over adjacent

00:09:34.900 --> 00:09:38.860
you go to the thirty degree angle because that's what we care about, tangent of thirty

00:09:38.860 --> 00:09:42.760
tangent of thirty opposite is two

00:09:42.760 --> 00:09:47.150
opposite is two and the adjacent is two square roots of three it's right next to it it's adjacent

00:09:47.150 --> 00:09:47.830
to it

00:09:47.830 --> 00:09:49.430
adjacent means next to

00:09:49.430 --> 00:09:51.720
so two square roots of three

00:09:51.720 --> 00:09:53.110
so this is equal to

00:09:53.110 --> 00:09:56.820
the twos cancel out one over the square root
of three

00:09:56.820 --> 00:10:00.340
or we could multiply the numerator and the denominator
by the square root of three

00:10:00.340 --> 00:10:01.740
so we have

00:10:01.740 --> 00:10:03.290
square root of three

00:10:03.290 --> 00:10:05.200
over square root of three

00:10:05.200 --> 00:10:09.600
and so this is going to be equal to the numerator
square root of three and then the denominator

00:10:09.600 --> 00:10:14.900
right over here is just going to be three so
thats we've rationalized a square root of three

00:10:14.900 --> 00:10:15.890
over three

00:10:15.890 --> 00:10:16.720
fair enough

00:10:16.720 --> 00:10:20.500
now lets use the same triangle to figure out the
trig ratios for the sixty degrees

00:10:20.500 --> 00:10:23.200
since we've already drawn it

00:10:23.200 --> 00:10:24.890
so what is

00:10:24.890 --> 00:10:30.580
what is in the sine of the sixty degrees and i think you're hopefully getting the hang of it now

00:10:30.580 --> 00:10:35.480
sine is opposite over adjacent. soh from the soh cah toa. from the sixty degree angle what side


00:10:35.480 --> 00:10:36.760
is opposite

00:10:36.760 --> 00:10:42.920
what opens out into the two square roots of three
so the opposite side is two square roots of three

00:10:42.920 --> 00:10:47.880
and from the sixty degree angle the adj-oh sorry its the

00:10:47.880 --> 00:10:54.420
opposite over hypotenuse, don't want to confuse you.

00:10:54.420 --> 00:10:58.750
so it is opposite over hypotenuse

00:10:58.750 --> 00:11:00.000
so it's two square roots of three over four. four is the hypotenuse.

00:11:00.000 --> 00:11:03.139
so it is equal to, this simplifies to square root of three over two.

00:11:03.139 --> 00:11:05.580
what is the cosine of sixty degrees. cosine of sixty degrees.

00:11:05.580 --> 00:11:10.330
so remember soh cah toa. cosine is adjacent over hypotenuse.

00:11:10.330 --> 00:11:15.070
adjacent is the two sides right next to the sixty degree angle so it's two

00:11:15.070 --> 00:11:17.920
over the hypotenuse which is four

00:11:17.920 --> 00:11:19.900
so this is equal to

00:11:19.900 --> 00:11:20.860
one-half

00:11:20.860 --> 00:11:22.120
and then finally

00:11:22.120 --> 00:11:24.460
what is the tangent, what is the tangent

00:11:26.000 --> 00:11:27.830
of sixty degrees

00:11:27.830 --> 00:11:32.790
well tangent soh cah toa tangent is opposite
over adjacent

00:11:32.790 --> 00:11:34.220
opposite the sixty degrees

00:11:34.220 --> 00:11:36.130
is two square roots of three

00:11:36.130 --> 00:11:37.940
two square roots of three

00:11:37.940 --> 00:11:39.570
and adjacent to that

00:11:39.570 --> 00:11:43.020
adjacent to that

00:11:43.020 --> 00:11:45.470
is two

00:11:45.470 --> 00:11:48.750
adjacent to sixty degrees is two

00:11:48.750 --> 00:11:52.630
so its opposite over adjacent

00:11:52.630 --> 00:11:56.000
two square roots of three over two which is just equal to

00:11:56.000 --> 00:11:58.150
the square root of three

00:11:58.150 --> 00:12:01.750
And I just wanted to - look how these are related

00:12:01.750 --> 00:12:03.365
the sine of thirty degrees is the same as the cosine of sixty degrees

00:12:03.365 --> 00:12:04.980
and then these guys are the inverse of each other and i think if you think a little bit about this triangle

00:12:05.440 --> 00:12:09.519
it will start to make sense why. we'll keep extending
this and give you a lot more practice in the next

00:12:09.519 --> 00:12:10.110
few videos