WEBVTT

00:00:01.106 --> 00:00:03.736 A:middle
>> Welcome to the Cypress
College math review

00:00:03.736 --> 00:00:04.976 A:middle
on trigonometric ratios.

00:00:12.166 --> 00:00:15.026 A:middle
There is a total of six
trigonometric ratios.

00:00:15.506 --> 00:00:19.906 A:middle
They are the sine,
cosine, tangent, cosecant,

00:00:20.216 --> 00:00:22.716 A:middle
secant and cotangent ratios.

00:00:23.376 --> 00:00:26.646 A:middle
But for geometry, we're
only talking about the sine,

00:00:26.646 --> 00:00:28.556 A:middle
cosine and tangent ratios.

00:00:30.116 --> 00:00:32.326 A:middle
You will learn about
the other ratios

00:00:32.326 --> 00:00:33.726 A:middle
when you take a trig class.

00:00:35.296 --> 00:00:38.396 A:middle
In the first objective,
we're going to use the sine,

00:00:38.456 --> 00:00:40.416 A:middle
cosine and tangent ratios

00:00:40.656 --> 00:00:41.966 A:middle
to determine side lengths
in right triangles.

00:00:46.356 --> 00:00:48.946 A:middle
Given the right triangle
ABC here,

00:00:50.256 --> 00:00:53.036 A:middle
segment AC is called
the hypotenuse

00:00:53.456 --> 00:00:56.246 A:middle
because it is the side
opposite from the right angle.

00:00:57.586 --> 00:01:01.496 A:middle
No matter how the right triangle
is positioned, the side opposite

00:01:01.496 --> 00:01:04.576 A:middle
from the right angle is
always the hypotenuse.

00:01:05.336 --> 00:01:07.736 A:middle
And, the other sides
are called the legs.

00:01:08.816 --> 00:01:12.416 A:middle
So, sides AB and BC
are called the legs.

00:01:12.996 --> 00:01:16.716 A:middle
And, they have lengths
of y and x, respectively,

00:01:17.206 --> 00:01:21.256 A:middle
while the hypotenuse AC
has a length of z units.

00:01:22.336 --> 00:01:30.016 A:middle
In reference to angle A,
segment AB is the side adjacent

00:01:30.016 --> 00:01:37.116 A:middle
to angle A. And, segment BC
is the side opposite angle A.

00:01:38.516 --> 00:01:43.206 A:middle
Notice that even though segment
AC is also a side adjacent

00:01:43.206 --> 00:01:46.976 A:middle
to angle A, however,
since it is opposite

00:01:46.976 --> 00:01:49.646 A:middle
from the right angle,
it is the hypotenuse.

00:01:50.316 --> 00:01:57.106 A:middle
We define sine of angle A to
be the ratio of the length

00:01:57.306 --> 00:02:01.916 A:middle
of the side opposite angle A and
the length of the hypotenuse.

00:02:06.166 --> 00:02:10.036 A:middle
Cosine of angle A is
the ratio of the length

00:02:10.156 --> 00:02:12.206 A:middle
of the side adjacent to angle A

00:02:12.206 --> 00:02:14.586 A:middle
and the length of
the hypotenuse.

00:02:16.616 --> 00:02:20.206 A:middle
A tangent of angle A is
the ratio of the length

00:02:20.206 --> 00:02:25.086 A:middle
of the side opposite angle A and
the length of the side adjacent

00:02:25.086 --> 00:02:34.716 A:middle
to angle A. We abbreviate sine
of angle A as sinA, but really,

00:02:34.716 --> 00:02:36.286 A:middle
we call this sine A. And,

00:02:37.926 --> 00:02:41.716 A:middle
the cosine of angle
A is written cosA.

00:02:41.906 --> 00:02:47.396 A:middle
The tangent of angle
A is written tanA.

00:02:47.776 --> 00:02:52.506 A:middle
However, when we read this,
we don't read it as tanA,

00:02:52.576 --> 00:02:53.976 A:middle
but we read it as tangent A.

00:02:58.046 --> 00:03:02.496 A:middle
To help us remember these
ratios, we can think of SOH,

00:03:03.616 --> 00:03:11.656 A:middle
CAH, TOA, where this
S stands for sine.

00:03:12.146 --> 00:03:13.916 A:middle
O is for opposite.

00:03:14.606 --> 00:03:16.196 A:middle
H is for hypotenuse.

00:03:17.236 --> 00:03:24.116 A:middle
CAH, C is for cosine, A is for
adjacent - the side adjacent

00:03:24.116 --> 00:03:26.956 A:middle
to the angle, H is
the hypotenuse.

00:03:27.856 --> 00:03:33.026 A:middle
TOA, T is for tangent,
O is opposite -

00:03:33.356 --> 00:03:37.816 A:middle
the side opposite the
angle, A is adjacent,

00:03:38.396 --> 00:03:41.536 A:middle
which is the side
adjacent to the angle.

00:03:43.596 --> 00:03:47.186 A:middle
We can also find the sine,
cosine and tangent ratios

00:03:47.186 --> 00:03:48.976 A:middle
for angle C using
the same formulas.

00:03:57.076 --> 00:03:59.976 A:middle
Let's practice applying the
three trig ratios for angle T.

00:04:06.146 --> 00:04:07.976 A:middle
Firstly, we identify
the hypotenuse.

00:04:09.586 --> 00:04:13.536 A:middle
We see that segment AT
is the side opposite

00:04:13.536 --> 00:04:16.596 A:middle
from the right angle R,
so it is the hypotenuse.

00:04:18.066 --> 00:04:21.646 A:middle
Segment TR is the side
adjacent to angle T

00:04:23.646 --> 00:04:30.226 A:middle
and segment AR is the side
opposite angle T. Sine

00:04:30.626 --> 00:04:40.366 A:middle
of angle T is defined as
opposite over hypotenuse.

00:04:41.036 --> 00:04:48.726 A:middle
It means the length of the
side opposite angle T divided

00:04:48.796 --> 00:04:50.536 A:middle
by the length of the hypotenuse.

00:04:52.346 --> 00:04:56.756 A:middle
And, we see that the side
opposite angle T has length 4

00:04:56.756 --> 00:05:02.766 A:middle
units and the hypotenuse has
length of 5 units, therefore,

00:05:03.246 --> 00:05:06.526 A:middle
sine of T is 4 over 5.

00:05:07.456 --> 00:05:16.226 A:middle
Cosine of T is defined as
adjacent over hypotenuse.

00:05:16.526 --> 00:05:20.876 A:middle
Remember that it is CAH.

00:05:26.076 --> 00:05:29.966 A:middle
The length of the side
adjacent to angle T is 3 units

00:05:30.536 --> 00:05:33.496 A:middle
and the length of the
hypotenuse is 5 units,

00:05:34.056 --> 00:05:36.706 A:middle
so cosine of T is 3 over 5.

00:05:39.306 --> 00:05:43.286 A:middle
Tangent of T is defined
as the length

00:05:43.286 --> 00:05:48.786 A:middle
of the side opposite angle
T divided by the length

00:05:48.786 --> 00:05:55.686 A:middle
of the side adjacent to angle
T. Or, remember that it is TOA.

00:05:58.056 --> 00:06:01.406 A:middle
Looking at the diagram, we
see that the side opposite

00:06:01.406 --> 00:06:06.346 A:middle
from angle T has length of 4
units, while the side adjacent

00:06:06.346 --> 00:06:10.176 A:middle
to angle T has length
of 3 units, so,

00:06:10.176 --> 00:06:12.976 A:middle
tangent of T is 4 over 3.

00:06:18.046 --> 00:06:19.976 A:middle
Pause the video and
try these problems.

00:06:31.046 --> 00:06:33.166 A:middle
In this example, we
will use a calculator

00:06:33.166 --> 00:06:35.076 A:middle
to find the trig ratios.

00:06:35.736 --> 00:06:38.746 A:middle
First, you need to make sure

00:06:38.746 --> 00:06:40.886 A:middle
that your calculator
is in degree mode.

00:06:42.376 --> 00:06:46.226 A:middle
So, on your calculator,
you click on this button

00:06:46.776 --> 00:06:50.226 A:middle
and then you want
to select degree.

00:06:50.226 --> 00:06:55.616 A:middle
You will know that your
calculator is in degree mode

00:06:55.806 --> 00:07:00.086 A:middle
if the bottom of
the screen has DEG.

00:07:00.476 --> 00:07:05.436 A:middle
Now, look on the calculator
to locate the button

00:07:05.436 --> 00:07:08.916 A:middle
for sine, cosine and tangent.

00:07:11.446 --> 00:07:14.866 A:middle
For part A, we want to
find sine of 35 degrees.

00:07:15.816 --> 00:07:17.206 A:middle
Then, on your calculator,

00:07:18.166 --> 00:07:22.166 A:middle
you would click sine
and press in 35.

00:07:23.446 --> 00:07:24.826 A:middle
And, this is the value.

00:07:26.266 --> 00:07:29.176 A:middle
We want to round it to
four decimal places,

00:07:30.046 --> 00:07:37.296 A:middle
so it is approximately 0.5736.

00:07:37.386 --> 00:07:39.956 A:middle
Try parts B and C yourself.

00:07:46.096 --> 00:07:53.616 A:middle
Did you get cosine of 52 degrees
to be approximately 0.6157?

00:07:54.166 --> 00:08:00.256 A:middle
And, tangent of 28
degrees is 0.5317.

00:08:01.146 --> 00:08:05.076 A:middle
What does it mean for
sine of 35 degrees

00:08:05.186 --> 00:08:08.346 A:middle
to be approximately 0.5736?

00:08:09.906 --> 00:08:16.006 A:middle
Well, it means that if we have
a right triangle where one

00:08:16.006 --> 00:08:21.496 A:middle
of the acute angles is 35
degrees and suppose the length

00:08:22.056 --> 00:08:26.566 A:middle
of the side opposite from the
35 degrees is y and the length

00:08:26.566 --> 00:08:33.566 A:middle
of the hypotenuse is some z
units, then the ratio of y

00:08:34.336 --> 00:08:42.406 A:middle
over z is approximately 0.5736.

00:08:43.876 --> 00:08:46.846 A:middle
We don't know exactly
the values of y and z,

00:08:47.216 --> 00:08:51.976 A:middle
but we know that
their ratio is 0.5736.

00:09:00.126 --> 00:09:04.386 A:middle
In this example, we want
to solve for x. Notice

00:09:04.386 --> 00:09:06.646 A:middle
that we cannot use the
Pythagorean theorem,

00:09:06.776 --> 00:09:09.266 A:middle
a squared plus b
squared equals c squared

00:09:09.726 --> 00:09:12.846 A:middle
because we do not know
the length of this side.

00:09:13.976 --> 00:09:16.406 A:middle
So, we will need
to use trigonometry

00:09:16.406 --> 00:09:16.976 A:middle
to find the value of x.

00:09:21.066 --> 00:09:23.416 A:middle
First, you need to
identify the hypotenuse.

00:09:25.166 --> 00:09:28.456 A:middle
This side is opposite
from the right angle,

00:09:28.916 --> 00:09:30.316 A:middle
so it is the hypotenuse.

00:09:31.046 --> 00:09:34.456 A:middle
This side has a length
of 14 units

00:09:34.666 --> 00:09:37.976 A:middle
and it is the side opposite
the 51 degrees angle.

00:09:46.086 --> 00:09:49.146 A:middle
Next, we decide which
trig ratio to use.

00:09:49.496 --> 00:09:53.816 A:middle
We have SOH, CAH, TOA.

00:09:55.806 --> 00:09:58.356 A:middle
And, here we have
opposite and hypotenuse,

00:09:59.066 --> 00:10:02.576 A:middle
so you see that we
should use the sine ratio.

00:10:04.676 --> 00:10:10.796 A:middle
So, we have the sine
of 51 degrees is equal

00:10:11.296 --> 00:10:16.556 A:middle
to opposite over hypotenuse.

00:10:16.736 --> 00:10:22.416 A:middle
The length of the side opposite

00:10:22.416 --> 00:10:27.976 A:middle
from the 51 degrees angle is 14
units and the hypotenuse is x.

00:10:33.046 --> 00:10:38.236 A:middle
Now, solve for x by multiplying
x by both sides of the equation,

00:10:38.786 --> 00:10:45.846 A:middle
then we get x times sine
of 51 is equal to 14,

00:10:46.406 --> 00:10:48.896 A:middle
since these two x's cancel.

00:10:53.126 --> 00:10:56.246 A:middle
Notice that sine 51 is just
a number that we can get

00:10:56.246 --> 00:11:01.426 A:middle
from the calculator so then we
can divide the equation by sine

00:11:01.426 --> 00:11:04.376 A:middle
of 51 to get x by itself.

00:11:04.926 --> 00:11:13.806 A:middle
So, we have x is equal
to 14 over sine of 51.

00:11:14.396 --> 00:11:17.976 A:middle
Try getting this value on
the calculator yourself.

00:11:22.366 --> 00:11:28.056 A:middle
You should get x is
approximately 18.01 units,

00:11:29.076 --> 00:11:32.166 A:middle
since we want to round the
answer to two decimal places.

00:11:34.146 --> 00:11:37.986 A:middle
X represents the length of the
hypotenuse, so it should be

00:11:37.986 --> 00:11:40.026 A:middle
in units, not degrees.

00:11:41.386 --> 00:11:45.606 A:middle
There is another way
we can solve for x,

00:11:46.666 --> 00:11:50.416 A:middle
that is we will use this
angle as the reference angle.

00:11:50.996 --> 00:11:55.216 A:middle
Since we know that the
sum of the measures

00:11:55.216 --> 00:11:58.776 A:middle
of the interior angles of
a triangle is 180 degrees,

00:11:59.286 --> 00:12:05.516 A:middle
then the measure of this angle
is 180 minus 51 minus 90,

00:12:05.826 --> 00:12:08.496 A:middle
since the right triangle
is 90 degrees.

00:12:09.196 --> 00:12:12.776 A:middle
And, this is equal to 39,
which means that the measure

00:12:12.776 --> 00:12:14.926 A:middle
of this angle is 39 degrees.

00:12:15.446 --> 00:12:19.716 A:middle
Relative to the 39
degrees angle,

00:12:20.346 --> 00:12:23.786 A:middle
this side is adjacent to it.

00:12:25.396 --> 00:12:29.676 A:middle
This side is the hypotenuse
because it is opposite

00:12:29.676 --> 00:12:33.296 A:middle
from the right angle,
even though it's adjacent

00:12:33.296 --> 00:12:37.756 A:middle
to the 39 degrees as well,
but because it is opposite

00:12:37.756 --> 00:12:40.196 A:middle
from the right angle, it
must be the hypotenuse.

00:12:41.886 --> 00:12:44.716 A:middle
Now, we determine which
trig ratio to use.

00:12:45.026 --> 00:12:48.736 A:middle
We have SOH, CAH, TOA.

00:12:49.436 --> 00:12:53.396 A:middle
And, since we have
adjacent and hypotenuse,

00:12:54.006 --> 00:12:57.286 A:middle
so the trig ratio we
should use is the cosine.

00:12:59.176 --> 00:13:04.296 A:middle
Therefore, we have
cosine of 39 is equal

00:13:04.296 --> 00:13:08.936 A:middle
to adjacent over hypotenuse.

00:13:16.046 --> 00:13:20.346 A:middle
The length of the side adjacent
to the 39 degrees is 14 units

00:13:20.746 --> 00:13:24.316 A:middle
and the hypotenuse has
a length of x units.

00:13:25.026 --> 00:13:31.316 A:middle
Now, we solve for x by multiply
both sides of the equation

00:13:31.556 --> 00:13:38.916 A:middle
by x. So, we have x times
cosine of 39 is equal to 14

00:13:38.916 --> 00:13:41.706 A:middle
since these two x's cancel out.

00:13:42.716 --> 00:13:46.176 A:middle
Cosine of 39 is just a
number that we can get

00:13:46.176 --> 00:13:50.006 A:middle
from a calculator so we
can divide both sides

00:13:50.006 --> 00:13:54.616 A:middle
of the equation by cosine
of 39 to get x by itself.

00:13:55.196 --> 00:14:03.356 A:middle
And, we have x is equal
to 14 over cosine of 39.

00:14:04.886 --> 00:14:06.976 A:middle
Try getting this number on
the calculator yourself.

00:14:12.226 --> 00:14:15.646 A:middle
Did you get 18.01?

00:14:16.186 --> 00:14:20.976 A:middle
This is the same number we got
using the sine ratio previously.

00:14:28.046 --> 00:14:34.516 A:middle
Part B. This right triangle has
an acute angle of 55 degrees,

00:14:35.496 --> 00:14:38.196 A:middle
so we will use it as
the reference angle.

00:14:39.976 --> 00:14:45.016 A:middle
This side is adjacent
to the 55 degrees angle.

00:14:45.576 --> 00:14:52.636 A:middle
And, this side is opposite
from the 55 degrees angle.

00:14:53.846 --> 00:15:00.206 A:middle
Next, we determine
which trig ratio to use.

00:15:00.596 --> 00:15:05.266 A:middle
We have SOH, CAH, TOA.

00:15:07.916 --> 00:15:11.466 A:middle
Looking on the diagram, we
have adjacent and opposite.

00:15:12.986 --> 00:15:19.256 A:middle
So, the trig ratio we should
use is tangent, therefore,

00:15:19.256 --> 00:15:23.836 A:middle
we have tangent of 55
degrees angle is equal

00:15:23.836 --> 00:15:28.416 A:middle
to opposite over adjacent.

00:15:28.466 --> 00:15:34.386 A:middle
When I say opposite
and adjacent,

00:15:34.386 --> 00:15:38.386 A:middle
I really mean the lengths of
the sides opposite or adjacent

00:15:38.386 --> 00:15:40.216 A:middle
to the specified angle.

00:15:41.836 --> 00:15:44.176 A:middle
So, the length of
the side opposite

00:15:44.176 --> 00:15:49.596 A:middle
from 55 degrees angle is
29 units, while the length

00:15:49.596 --> 00:15:51.976 A:middle
of the side adjacent to
the 55 degrees angle is x.

00:15:58.066 --> 00:16:03.316 A:middle
We solve for x by multiplying
both sides of the equation by x

00:16:04.436 --> 00:16:10.596 A:middle
and we get x times tangent
of 55 is equal to 29.

00:16:11.146 --> 00:16:15.536 A:middle
And, divide both
sides by tangent 55,

00:16:15.806 --> 00:16:21.056 A:middle
since it's just a number that
we can get from a calculator.

00:16:21.116 --> 00:16:26.836 A:middle
So, x is equal to 29
over tangent of 55.

00:16:26.836 --> 00:16:30.896 A:middle
Try and get this number on
the calculator yourself.

00:16:35.296 --> 00:16:38.976 A:middle
Did you get approximately
20.31 units?

00:16:49.046 --> 00:16:51.846 A:middle
As part A, there is
another way to solve for x

00:16:52.086 --> 00:16:54.976 A:middle
by using the other acute
angle as the reference angle.

00:17:07.046 --> 00:17:14.206 A:middle
The measure of this angle is
equal to 180 minus 55 minus 90.

00:17:14.766 --> 00:17:18.366 A:middle
So, that is equal to 35.

00:17:19.186 --> 00:17:21.916 A:middle
So, the measure of this
angle is 35 degrees.

00:17:22.456 --> 00:17:30.476 A:middle
Relative to the 35 degrees
angle, this side is adjacent

00:17:31.326 --> 00:17:34.886 A:middle
to it and this side is opposite.

00:17:42.056 --> 00:17:45.136 A:middle
Since the sides that we have
in the triangle are opposite

00:17:45.136 --> 00:17:48.576 A:middle
and adjacent, then
we will use tangent.

00:17:49.336 --> 00:17:53.586 A:middle
So, we have tangent
of 35 degrees is equal

00:17:53.616 --> 00:17:58.826 A:middle
to opposite over adjacent.

00:18:04.066 --> 00:18:06.096 A:middle
So, the length of
the side opposite

00:18:06.096 --> 00:18:09.756 A:middle
from the 35 degrees
angle is x and the length

00:18:09.756 --> 00:18:13.946 A:middle
of the side adjacent to the
35 degrees angle is 29 units.

00:18:19.046 --> 00:18:21.736 A:middle
To solve for x, we will
multiply both sides

00:18:21.736 --> 00:18:22.976 A:middle
of the equation by 29.

00:18:28.046 --> 00:18:36.056 A:middle
And, we get 29 times tangent
of 35 is equal to x. Try

00:18:36.266 --> 00:18:39.556 A:middle
to get this value on
the calculator yourself.

00:18:40.146 --> 00:18:43.726 A:middle
Did you get 20.31?

00:18:44.296 --> 00:18:48.566 A:middle
And, you see that
this value is the same

00:18:48.566 --> 00:18:50.976 A:middle
as the value we got
using the other way.

00:19:00.116 --> 00:19:03.296 A:middle
Part C. Just like the
two previous examples,

00:19:03.746 --> 00:19:07.856 A:middle
we will first identify
the sides.

00:19:07.856 --> 00:19:10.256 A:middle
Relative to the 47
degrees angle,

00:19:10.886 --> 00:19:11.976 A:middle
this side is adjacent to it.

00:19:18.066 --> 00:19:19.736 A:middle
And, this side is the hypotenuse

00:19:20.416 --> 00:19:22.936 A:middle
because it is opposite
the right angle.

00:19:24.446 --> 00:19:27.026 A:middle
Now, we decide which
trig ratio to use.

00:19:28.496 --> 00:19:31.096 A:middle
It helps if you write
down SOH, CAH, TOA.

00:19:31.646 --> 00:19:37.316 A:middle
We have adjacent and
hypotenuse, therefor,

00:19:37.316 --> 00:19:40.056 A:middle
we will use the cosine ratio.

00:19:40.606 --> 00:19:46.056 A:middle
So, we have cosine
of 47 is equal

00:19:46.056 --> 00:19:50.946 A:middle
to adjacent over hypotenuse.

00:19:56.126 --> 00:20:01.266 A:middle
Where the length of the adjacent
side is 28 units and the length

00:20:01.266 --> 00:20:03.046 A:middle
of the hypotenuse is x units.

00:20:04.196 --> 00:20:09.486 A:middle
We multiply the equation by x

00:20:10.476 --> 00:20:18.826 A:middle
and we get x cosine
of 47 is equal to 28.

00:20:19.296 --> 00:20:27.496 A:middle
Now, divide the equation by
cosine of 47 to get x by itself.

00:20:29.106 --> 00:20:36.246 A:middle
We get x is equal to
28 over cosine of 47,

00:20:37.336 --> 00:20:40.976 A:middle
which is approximately
41.06 units.

00:20:46.326 --> 00:20:49.046 A:middle
We can also solve for
x by using this angle

00:20:49.086 --> 00:20:49.976 A:middle
as the reference angle.

00:21:06.126 --> 00:21:12.246 A:middle
This angle is then 180 minus
90 minus 47, which is 43.

00:21:13.246 --> 00:21:18.696 A:middle
So, the angle is 43
degrees, therefore,

00:21:18.846 --> 00:21:22.106 A:middle
this side is opposite
the 43 degrees angle.

00:21:22.666 --> 00:21:26.416 A:middle
This side is still
the hypotenuse

00:21:26.476 --> 00:21:27.916 A:middle
because it is opposite
the right angle.

00:21:34.496 --> 00:21:37.586 A:middle
And, you see that we
should use the sine ratio

00:21:37.586 --> 00:21:41.876 A:middle
because we have opposite
and hypotenuse, therefore,

00:21:41.876 --> 00:21:48.316 A:middle
we have sine of 43
degrees is equal to 28

00:21:48.416 --> 00:21:52.496 A:middle
over x. Multiply
x by both sides.

00:21:53.316 --> 00:22:00.376 A:middle
We get x times sine
of 43 is equal to 28.

00:22:02.066 --> 00:22:05.966 A:middle
And, divide both sides by
sine of 43 to get x by itself.

00:22:11.046 --> 00:22:16.116 A:middle
So, we have x is equal
to 28 over sine of 43.

00:22:17.936 --> 00:22:20.456 A:middle
Try getting this value
on the calculator.

00:22:22.926 --> 00:22:25.626 A:middle
You should get the same
value as we did on the left,

00:22:25.916 --> 00:22:30.766 A:middle
which is approximately
41.06 units.

00:22:37.046 --> 00:22:38.976 A:middle
Pause the video and
try these problems.

00:22:50.286 --> 00:22:54.246 A:middle
You just learned to use the
trig ratios to find the length

00:22:54.246 --> 00:22:55.946 A:middle
of the sides of triangles.

00:22:57.046 --> 00:23:00.706 A:middle
In this objective, you are
going to learn to use the sine,

00:23:00.706 --> 00:23:02.546 A:middle
cosine and tangent ratios

00:23:02.956 --> 00:23:05.916 A:middle
to determine angle
measures in right triangles.

00:23:06.566 --> 00:23:11.596 A:middle
If we know the sine, cosine
or tangent ratio for an angle,

00:23:12.286 --> 00:23:15.556 A:middle
we can use an inverse to find
the measure of the angle.

00:23:17.006 --> 00:23:20.496 A:middle
Sine to the negative 1
is read sine inverse.

00:23:21.436 --> 00:23:25.696 A:middle
Cosine to the negative 1
is read cosine inverse.

00:23:26.656 --> 00:23:28.556 A:middle
And, this is read
tangent inverse.

00:23:31.476 --> 00:23:34.306 A:middle
In this example, we
want to use a calculator

00:23:34.306 --> 00:23:36.556 A:middle
to approximate the
measure of angle A

00:23:36.586 --> 00:23:38.736 A:middle
to the nearest whole degree.

00:23:39.296 --> 00:23:46.776 A:middle
Part A, cosine A
is equal to 0.6052.

00:23:47.386 --> 00:23:53.986 A:middle
This is asking cosine
of what angle is 0.6052?

00:23:55.276 --> 00:23:57.426 A:middle
So, to get A by itself,
we do this.

00:23:58.626 --> 00:24:07.796 A:middle
We have A is equal to
cosine inverse of 0.6052.

00:24:07.926 --> 00:24:09.886 A:middle
Now, we press this
on a calculator.

00:24:10.576 --> 00:24:16.006 A:middle
On your calculator, you see
on top of the cosine button,

00:24:16.126 --> 00:24:22.376 A:middle
there's a cosine inverse and
to get to cosine inverse,

00:24:22.846 --> 00:24:24.706 A:middle
you need to press second.

00:24:25.956 --> 00:24:31.186 A:middle
So, on your calculator, press
second then this cosine button

00:24:31.366 --> 00:24:34.636 A:middle
and then press in
the decimal 0.6052.

00:24:36.056 --> 00:24:41.986 A:middle
You should get 52.7567649.

00:24:42.306 --> 00:24:44.226 A:middle
Since the question
wants us to round

00:24:44.226 --> 00:24:47.166 A:middle
to the nearest whole degree,
that means we want to round

00:24:47.166 --> 00:24:52.276 A:middle
to the nearest whole number,
then this is approximately 53.

00:24:53.346 --> 00:25:00.036 A:middle
So, we say that the measure of
angle A is about 53 degrees.

00:25:02.806 --> 00:25:04.866 A:middle
What does the answer
in part A mean?

00:25:06.566 --> 00:25:11.726 A:middle
Well, it means that if we have
a right triangle where one

00:25:11.726 --> 00:25:18.586 A:middle
of the acute angle's we call
it A. And, if the side adjacent

00:25:18.586 --> 00:25:25.666 A:middle
to angle A has some y units and
the hypotenuse has some z units,

00:25:26.426 --> 00:25:36.206 A:middle
then it's saying that if y
over z is equal to 0.6052,

00:25:36.816 --> 00:25:43.076 A:middle
then the measure of angle
A is about 53 degrees.

00:25:44.366 --> 00:25:47.766 A:middle
That means, this angle
right here is 53 degrees.

00:25:50.496 --> 00:25:54.426 A:middle
We don't know exactly what
the values of y and z are,

00:25:55.556 --> 00:25:59.796 A:middle
but if the ratio is .6052,

00:26:00.096 --> 00:26:02.946 A:middle
the measure of angle A
must be about 53 degrees.

00:26:07.166 --> 00:26:11.916 A:middle
Part B. Sine A is
equal to .2318.

00:26:12.946 --> 00:26:22.766 A:middle
Once again, this is asking
sine of what angle is .2318?

00:26:22.766 --> 00:26:29.346 A:middle
So, A is equal to sine
inverse of 0.2318.

00:26:31.686 --> 00:26:37.086 A:middle
On your calculator,
press second, then sine,

00:26:37.836 --> 00:26:41.096 A:middle
then press in the decimal .2318.

00:26:42.416 --> 00:26:48.936 A:middle
You should get this value and
it is rounded to 13, therefore,

00:26:49.206 --> 00:26:52.976 A:middle
the measure of angle A is
approximately 13 degrees.

00:26:57.156 --> 00:26:57.976 A:middle
Try part C yourself.

00:27:04.046 --> 00:27:11.156 A:middle
You should get A is equal to
tangent inverse of 0.8325,

00:27:12.116 --> 00:27:14.856 A:middle
which is approximately 40.

00:27:15.916 --> 00:27:18.976 A:middle
So, the measure of angle
A is about 40 degrees.

00:27:25.046 --> 00:27:28.396 A:middle
In conclusion, whenever you want
to find the measure of an angle,

00:27:28.746 --> 00:27:32.666 A:middle
then you need to use the
inverse of which ever trig ratio

00:27:32.756 --> 00:27:33.976 A:middle
that is applicable
to the problem.

00:27:42.066 --> 00:27:45.576 A:middle
In this example, we want to
find the measure of angle H,

00:27:45.936 --> 00:27:47.426 A:middle
rounded to the nearest degree.

00:27:49.476 --> 00:27:54.756 A:middle
This is angle H and the two
known sides are HB and HX.

00:27:55.336 --> 00:28:01.976 A:middle
Segment HB is adjacent
to angle H.

00:28:08.086 --> 00:28:11.666 A:middle
And, segment HX is the side
opposite the right angle,

00:28:11.906 --> 00:28:12.826 A:middle
so it is the hypotenuse.

00:28:17.306 --> 00:28:20.166 A:middle
Now, we determine which
trig ratio to use.

00:28:20.706 --> 00:28:22.976 A:middle
We have SOH, CAH, TOA.

00:28:28.296 --> 00:28:30.466 A:middle
Pause the video and determine

00:28:30.686 --> 00:28:32.976 A:middle
which trig ratio we
should use for this one.

00:28:38.046 --> 00:28:41.016 A:middle
You see that we should
use the cosine ratio

00:28:42.036 --> 00:28:44.566 A:middle
because we have the
adjacent and the hypotenuse.

00:28:45.376 --> 00:28:48.006 A:middle
A is for adjacent.

00:28:48.236 --> 00:28:49.676 A:middle
H is for the hypotenuse.

00:28:50.386 --> 00:28:58.906 A:middle
So, we have cosine of angle H is
equal to adjacent, which is 6,

00:28:59.266 --> 00:29:01.486 A:middle
over hypotenuse, which is 10.

00:29:02.066 --> 00:29:09.296 A:middle
So, we have H is equal to
cosine inverse of 6 over 10.

00:29:10.956 --> 00:29:12.456 A:middle
Press this on the calculator.

00:29:13.196 --> 00:29:18.076 A:middle
You should press second, cosine,
and then 6 divided by 10.

00:29:19.596 --> 00:29:22.796 A:middle
It should give you 53.1.

00:29:24.466 --> 00:29:26.976 A:middle
We want to round this to
the nearest whole number,

00:29:26.976 --> 00:29:29.446 A:middle
so it should be 53, therefore,

00:29:29.546 --> 00:29:32.896 A:middle
the measure of angle
H is about 53 degrees.

00:29:40.116 --> 00:29:45.166 A:middle
In this example, we want to find
the measure of angle M. We see

00:29:45.446 --> 00:29:48.586 A:middle
that segment AT is the
side opposite angle M

00:29:49.146 --> 00:29:55.636 A:middle
and segment MA is
the hypotenuse,

00:29:57.116 --> 00:30:00.356 A:middle
because it is opposite
the right angle.

00:30:01.856 --> 00:30:04.936 A:middle
Since we have the opposite
side and the hypotenuse,

00:30:05.486 --> 00:30:06.726 A:middle
then we should use sine.

00:30:08.146 --> 00:30:11.216 A:middle
So, opposite and hypotenuse.

00:30:11.906 --> 00:30:18.986 A:middle
So, we have the sine of angle
M is equal to the length

00:30:18.986 --> 00:30:23.946 A:middle
of the opposite side, which
is 16, divided by the length

00:30:23.946 --> 00:30:26.146 A:middle
of the hypotenuse, which is 25.

00:30:28.326 --> 00:30:34.666 A:middle
So, M is equal to sine
inverse of 16 over 25.

00:30:35.216 --> 00:30:38.976 A:middle
Press this on the
calculator yourself.

00:30:43.446 --> 00:30:46.046 A:middle
Did you get 39.8?

00:30:48.416 --> 00:30:52.426 A:middle
This is rounded to 40 degrees

00:30:53.456 --> 00:30:54.926 A:middle
and that is the measure
of angle M.

00:31:02.486 --> 00:31:05.446 A:middle
In this next example, we
want to find the measure

00:31:05.826 --> 00:31:13.046 A:middle
of angle A. Segment AB is
the side adjacent to angle A

00:31:15.216 --> 00:31:24.466 A:middle
and segment BC is the
side opposite angle A. So,

00:31:24.846 --> 00:31:28.126 A:middle
you see that we should
use tangent, TOA.

00:31:28.926 --> 00:31:33.766 A:middle
O for opposite and
A for adjacent.

00:31:36.146 --> 00:31:42.706 A:middle
So, we have tangent of angle
A is equal to opposite,

00:31:42.706 --> 00:31:46.026 A:middle
which is 12, over
adjacent, which is 7.

00:31:47.546 --> 00:31:54.256 A:middle
So, A is equal to tangent
inverse of 12 over 7.

00:31:55.336 --> 00:31:57.286 A:middle
Press this in the
calculator yourself.

00:31:57.826 --> 00:32:02.396 A:middle
Did you get 59.7?

00:32:02.936 --> 00:32:07.796 A:middle
So, this is rounded
to 60 degrees

00:32:08.436 --> 00:32:09.896 A:middle
and that is the measure
of angle A.

00:32:15.266 --> 00:32:17.976 A:middle
Pause the video and
try these problems.