>> Welcome to the Cypress College math review
on trigonometric ratios.
There is a total of six trigonometric ratios.
They are the sine, cosine, tangent, cosecant,
secant and cotangent ratios.
But for geometry, we're only talking about the sine,
cosine and tangent ratios.
You will learn about the other ratios
when you take a trig class.
In the first objective, we're going to use the sine,
cosine and tangent ratios
to determine side lengths in right triangles.
Given the right triangle ABC here,
segment AC is called the hypotenuse
because it is the side opposite from the right angle.
No matter how the right triangle is positioned, the side opposite
from the right angle is always the hypotenuse.
And, the other sides are called the legs.
So, sides AB and BC are called the legs.
And, they have lengths of y and x, respectively,
while the hypotenuse AC has a length of z units.
In reference to angle A, segment AB is the side adjacent
to angle A. And, segment BC is the side opposite angle A.
Notice that even though segment AC is also a side adjacent
to angle A, however, since it is opposite
from the right angle, it is the hypotenuse.
We define sine of angle A to be the ratio of the length
of the side opposite angle A and the length of the hypotenuse.
Cosine of angle A is the ratio of the length
of the side adjacent to angle A
and the length of the hypotenuse.
A tangent of angle A is the ratio of the length
of the side opposite angle A and the length of the side adjacent
to angle A. We abbreviate sine of angle A as sinA, but really,
we call this sine A. And,
the cosine of angle A is written cosA.
The tangent of angle A is written tanA.
However, when we read this, we don't read it as tanA,
but we read it as tangent A.
To help us remember these ratios, we can think of SOH,
CAH, TOA, where this S stands for sine.
O is for opposite.
H is for hypotenuse.
CAH, C is for cosine, A is for adjacent - the side adjacent
to the angle, H is the hypotenuse.
TOA, T is for tangent, O is opposite -
the side opposite the angle, A is adjacent,
which is the side adjacent to the angle.
We can also find the sine, cosine and tangent ratios
for angle C using the same formulas.
Let's practice applying the three trig ratios for angle T.
Firstly, we identify the hypotenuse.
We see that segment AT is the side opposite
from the right angle R, so it is the hypotenuse.
Segment TR is the side adjacent to angle T
and segment AR is the side opposite angle T. Sine
of angle T is defined as opposite over hypotenuse.
It means the length of the side opposite angle T divided
by the length of the hypotenuse.
And, we see that the side opposite angle T has length 4
units and the hypotenuse has length of 5 units, therefore,
sine of T is 4 over 5.
Cosine of T is defined as adjacent over hypotenuse.
Remember that it is CAH.
The length of the side adjacent to angle T is 3 units
and the length of the hypotenuse is 5 units,
so cosine of T is 3 over 5.
Tangent of T is defined as the length
of the side opposite angle T divided by the length
of the side adjacent to angle T. Or, remember that it is TOA.
Looking at the diagram, we see that the side opposite
from angle T has length of 4 units, while the side adjacent
to angle T has length of 3 units, so,
tangent of T is 4 over 3.
Pause the video and try these problems.
In this example, we will use a calculator
to find the trig ratios.
First, you need to make sure
that your calculator is in degree mode.
So, on your calculator, you click on this button
and then you want to select degree.
You will know that your calculator is in degree mode
if the bottom of the screen has DEG.
Now, look on the calculator to locate the button
for sine, cosine and tangent.
For part A, we want to find sine of 35 degrees.
Then, on your calculator,
you would click sine and press in 35.
And, this is the value.
We want to round it to four decimal places,
so it is approximately 0.5736.
Try parts B and C yourself.
Did you get cosine of 52 degrees to be approximately 0.6157?
And, tangent of 28 degrees is 0.5317.
What does it mean for sine of 35 degrees
to be approximately 0.5736?
Well, it means that if we have a right triangle where one
of the acute angles is 35 degrees and suppose the length
of the side opposite from the 35 degrees is y and the length
of the hypotenuse is some z units, then the ratio of y
over z is approximately 0.5736.
We don't know exactly the values of y and z,
but we know that their ratio is 0.5736.
In this example, we want to solve for x. Notice
that we cannot use the Pythagorean theorem,
a squared plus b squared equals c squared
because we do not know the length of this side.
So, we will need to use trigonometry
to find the value of x.
First, you need to identify the hypotenuse.
This side is opposite from the right angle,
so it is the hypotenuse.
This side has a length of 14 units
and it is the side opposite the 51 degrees angle.
Next, we decide which trig ratio to use.
We have SOH, CAH, TOA.
And, here we have opposite and hypotenuse,
so you see that we should use the sine ratio.
So, we have the sine of 51 degrees is equal
to opposite over hypotenuse.
The length of the side opposite
from the 51 degrees angle is 14 units and the hypotenuse is x.
Now, solve for x by multiplying x by both sides of the equation,
then we get x times sine of 51 is equal to 14,
since these two x's cancel.
Notice that sine 51 is just a number that we can get
from the calculator so then we can divide the equation by sine
of 51 to get x by itself.
So, we have x is equal to 14 over sine of 51.
Try getting this value on the calculator yourself.
You should get x is approximately 18.01 units,
since we want to round the answer to two decimal places.
X represents the length of the hypotenuse, so it should be
in units, not degrees.
There is another way we can solve for x,
that is we will use this angle as the reference angle.
Since we know that the sum of the measures
of the interior angles of a triangle is 180 degrees,
then the measure of this angle is 180 minus 51 minus 90,
since the right triangle is 90 degrees.
And, this is equal to 39, which means that the measure
of this angle is 39 degrees.
Relative to the 39 degrees angle,
this side is adjacent to it.
This side is the hypotenuse because it is opposite
from the right angle, even though it's adjacent
to the 39 degrees as well, but because it is opposite
from the right angle, it must be the hypotenuse.
Now, we determine which trig ratio to use.
We have SOH, CAH, TOA.
And, since we have adjacent and hypotenuse,
so the trig ratio we should use is the cosine.
Therefore, we have cosine of 39 is equal
to adjacent over hypotenuse.
The length of the side adjacent to the 39 degrees is 14 units
and the hypotenuse has a length of x units.
Now, we solve for x by multiply both sides of the equation
by x. So, we have x times cosine of 39 is equal to 14
since these two x's cancel out.
Cosine of 39 is just a number that we can get
from a calculator so we can divide both sides
of the equation by cosine of 39 to get x by itself.
And, we have x is equal to 14 over cosine of 39.
Try getting this number on the calculator yourself.
Did you get 18.01?
This is the same number we got using the sine ratio previously.
Part B. This right triangle has an acute angle of 55 degrees,
so we will use it as the reference angle.
This side is adjacent to the 55 degrees angle.
And, this side is opposite from the 55 degrees angle.
Next, we determine which trig ratio to use.
We have SOH, CAH, TOA.
Looking on the diagram, we have adjacent and opposite.
So, the trig ratio we should use is tangent, therefore,
we have tangent of 55 degrees angle is equal
to opposite over adjacent.
When I say opposite and adjacent,
I really mean the lengths of the sides opposite or adjacent
to the specified angle.
So, the length of the side opposite
from 55 degrees angle is 29 units, while the length
of the side adjacent to the 55 degrees angle is x.
We solve for x by multiplying both sides of the equation by x
and we get x times tangent of 55 is equal to 29.
And, divide both sides by tangent 55,
since it's just a number that we can get from a calculator.
So, x is equal to 29 over tangent of 55.
Try and get this number on the calculator yourself.
Did you get approximately 20.31 units?
As part A, there is another way to solve for x
by using the other acute angle as the reference angle.
The measure of this angle is equal to 180 minus 55 minus 90.
So, that is equal to 35.
So, the measure of this angle is 35 degrees.
Relative to the 35 degrees angle, this side is adjacent
to it and this side is opposite.
Since the sides that we have in the triangle are opposite
and adjacent, then we will use tangent.
So, we have tangent of 35 degrees is equal
to opposite over adjacent.
So, the length of the side opposite
from the 35 degrees angle is x and the length
of the side adjacent to the 35 degrees angle is 29 units.
To solve for x, we will multiply both sides
of the equation by 29.
And, we get 29 times tangent of 35 is equal to x. Try
to get this value on the calculator yourself.
Did you get 20.31?
And, you see that this value is the same
as the value we got using the other way.
Part C. Just like the two previous examples,
we will first identify the sides.
Relative to the 47 degrees angle,
this side is adjacent to it.
And, this side is the hypotenuse
because it is opposite the right angle.
Now, we decide which trig ratio to use.
It helps if you write down SOH, CAH, TOA.
We have adjacent and hypotenuse, therefor,
we will use the cosine ratio.
So, we have cosine of 47 is equal
to adjacent over hypotenuse.
Where the length of the adjacent side is 28 units and the length
of the hypotenuse is x units.
We multiply the equation by x
and we get x cosine of 47 is equal to 28.
Now, divide the equation by cosine of 47 to get x by itself.
We get x is equal to 28 over cosine of 47,
which is approximately 41.06 units.
We can also solve for x by using this angle
as the reference angle.
This angle is then 180 minus 90 minus 47, which is 43.
So, the angle is 43 degrees, therefore,
this side is opposite the 43 degrees angle.
This side is still the hypotenuse
because it is opposite the right angle.
And, you see that we should use the sine ratio
because we have opposite and hypotenuse, therefore,
we have sine of 43 degrees is equal to 28
over x. Multiply x by both sides.
We get x times sine of 43 is equal to 28.
And, divide both sides by sine of 43 to get x by itself.
So, we have x is equal to 28 over sine of 43.
Try getting this value on the calculator.
You should get the same value as we did on the left,
which is approximately 41.06 units.
Pause the video and try these problems.
You just learned to use the trig ratios to find the length
of the sides of triangles.
In this objective, you are going to learn to use the sine,
cosine and tangent ratios
to determine angle measures in right triangles.
If we know the sine, cosine or tangent ratio for an angle,
we can use an inverse to find the measure of the angle.
Sine to the negative 1 is read sine inverse.
Cosine to the negative 1 is read cosine inverse.
And, this is read tangent inverse.
In this example, we want to use a calculator
to approximate the measure of angle A
to the nearest whole degree.
Part A, cosine A is equal to 0.6052.
This is asking cosine of what angle is 0.6052?
So, to get A by itself, we do this.
We have A is equal to cosine inverse of 0.6052.
Now, we press this on a calculator.
On your calculator, you see on top of the cosine button,
there's a cosine inverse and to get to cosine inverse,
you need to press second.
So, on your calculator, press second then this cosine button
and then press in the decimal 0.6052.
You should get 52.7567649.
Since the question wants us to round
to the nearest whole degree, that means we want to round
to the nearest whole number, then this is approximately 53.
So, we say that the measure of angle A is about 53 degrees.
What does the answer in part A mean?
Well, it means that if we have a right triangle where one
of the acute angle's we call it A. And, if the side adjacent
to angle A has some y units and the hypotenuse has some z units,
then it's saying that if y over z is equal to 0.6052,
then the measure of angle A is about 53 degrees.
That means, this angle right here is 53 degrees.
We don't know exactly what the values of y and z are,
but if the ratio is .6052,
the measure of angle A must be about 53 degrees.
Part B. Sine A is equal to .2318.
Once again, this is asking sine of what angle is .2318?
So, A is equal to sine inverse of 0.2318.
On your calculator, press second, then sine,
then press in the decimal .2318.
You should get this value and it is rounded to 13, therefore,
the measure of angle A is approximately 13 degrees.
Try part C yourself.
You should get A is equal to tangent inverse of 0.8325,
which is approximately 40.
So, the measure of angle A is about 40 degrees.
In conclusion, whenever you want to find the measure of an angle,
then you need to use the inverse of which ever trig ratio
that is applicable to the problem.
In this example, we want to find the measure of angle H,
rounded to the nearest degree.
This is angle H and the two known sides are HB and HX.
Segment HB is adjacent to angle H.
And, segment HX is the side opposite the right angle,
so it is the hypotenuse.
Now, we determine which trig ratio to use.
We have SOH, CAH, TOA.
Pause the video and determine
which trig ratio we should use for this one.
You see that we should use the cosine ratio
because we have the adjacent and the hypotenuse.
A is for adjacent.
H is for the hypotenuse.
So, we have cosine of angle H is equal to adjacent, which is 6,
over hypotenuse, which is 10.
So, we have H is equal to cosine inverse of 6 over 10.
Press this on the calculator.
You should press second, cosine, and then 6 divided by 10.
It should give you 53.1.
We want to round this to the nearest whole number,
so it should be 53, therefore,
the measure of angle H is about 53 degrees.
In this example, we want to find the measure of angle M. We see
that segment AT is the side opposite angle M
and segment MA is the hypotenuse,
because it is opposite the right angle.
Since we have the opposite side and the hypotenuse,
then we should use sine.
So, opposite and hypotenuse.
So, we have the sine of angle M is equal to the length
of the opposite side, which is 16, divided by the length
of the hypotenuse, which is 25.
So, M is equal to sine inverse of 16 over 25.
Press this on the calculator yourself.
Did you get 39.8?
This is rounded to 40 degrees
and that is the measure of angle M.
In this next example, we want to find the measure
of angle A. Segment AB is the side adjacent to angle A
and segment BC is the side opposite angle A. So,
you see that we should use tangent, TOA.
O for opposite and A for adjacent.
So, we have tangent of angle A is equal to opposite,
which is 12, over adjacent, which is 7.
So, A is equal to tangent inverse of 12 over 7.
Press this in the calculator yourself.
Did you get 59.7?
So, this is rounded to 60 degrees
and that is the measure of angle A.
Pause the video and try these problems.