Hello everyone, welcome back to another video.
In the last video, we did the important ratios and values
of trigonometry related to height and distance.
Then I gave you the homework to write the values of sin,
cos and tan.
And I gave you the homework for cosec, sec and cot theta.
I hope you all have done it.
And if you have done it, then you can match it from here.
Note it down once or take a screenshot of it.
This will be your table for all the six trigonometric
ratios.
Then we did Pythagoras theorem.
So you must have heard what I told you in Pythagoras
theorem.
Hypotenuse square is equal to square of sum of other two
sides.
If we square of other two sides and if we do their sum,
then it will be equal to hypotenuse square.
So with the help of this, we will get trigonometric
identities.
Now what are these trigonometric identities?
We have done the ratios, trigonometric ratios, sin, cos,
tan.
With the help of these, we will get trigonometric
identities.
What will be the meaning of identity?
It will be 1.
Whatever equation will come, 1 will definitely come.
So let us suppose you have Pythagoras theorem.
Hypotenuse square is equal to base square plus
perpendicular square.
Now if we put their values.
Don't put values.
What you do is, divide the whole equation by hypotenuse
square.
Dividing the whole equation by hypotenuse square.
Divide by hypotenuse square.
So if we divide hypotenuse square by hypotenuse square,
then we will get 1.
If we get 1, then what will happen?
We call it identity 1.
So from here only we will get trigonometric identities.
Now if we divide base square by hypotenuse square.
Plus perpendicular square.
How do we divide hypotenuse square?
Now what do we do with this square? We take this square
together.
Base upon whole square will be this.
Plus perpendicular upon hypotenuse will be whole square.
And this was 1.
So base upon hypotenuse.
What did we have base upon hypotenuse? Base upon hypotenuse
was whose formula?
Sin theta.
Now what are we doing for theta? So we will get sin theta
whole square.
Plus perpendicular hypotenuse was whose formula? Cos theta.
So cos theta whole square.
Now write cos theta whole square or sin theta whole square.
We can also write it like this.
Sin and square theta.
Both means the same thing.
That what is your sin theta?
It is multiplied twice.
That's it.
Plus cos square theta.
Whose identity is this?
Identity that is 1.
Now in this trigonometric ratio is also being used.
What is I is equal to 1?
So we will call this as trigonometric identity.
This is one of our trigonometric identities.
That is sin square theta plus cos square theta is equal to
1.
This is our first trigonometric identity.
Trigonometric identity.
Now how will you find the rest of the identities?
What will you do with the rest of the identities?
We have to make 1 only in this equation.
Now how did we make 1 once? The whole equation.
By dividing it by hypotenuse square, this side becomes 1.
Now if we have to make this base square term 1.
So what we will do?
Divide the whole equation by base square.
Similarly how will we make one more identity?
When we will divide the whole equation by perpendicular
square.
So what will you get by doing this?
You will get three trigonometric identities.
You can also find the rest of the identities with the help
of this.
What can you do?
Divide the whole equation by sin square theta.
So what will you get? 1 upon sin square theta that is cos
square theta is equal to sin square divided by sin square.
1 plus cos square divided by sin square.
That is cos upon sin whole square.
What will that become?
Cot square theta.
So what will you get?
Cos square theta is equal to 1 plus cot square theta.
This is your second identity.
Second trigonometric identity.
How will the third one be? Divide the whole equation by cos
square.
So what will this become? 1 upon cos square that is sec
square theta.
So sec square theta is equal to sin square upon cos square.
And that is sin upon cos whole square that is tan square
theta.
And plus cos upon cos square that is 1.
So this is your sec square theta is equal to tan square
plus 1.
This is your third trigonometric identity.
So this is your first, second and third three trigonometric
identities.
Mainly you can remember this.
With the help of this, you can find the rest of the
identities.
The rest of the identities will also be found.
So we have done this trigonometric identities.
Next is the one on which questions will be made of height
and distance.
Right? Till now we have read these tools which will be used
in those questions. Right? What we will do next, the main
thing is that is angle of elevation and angle of depression.
So let us write the heading.
Angle of elevation and next is angle of depression.
Now what happens is angle of elevation and angle of
depression.
From the angle, we will know that we will talk about the
angle only.
Let us suppose I am standing here.
And what I am doing is that I am looking at the tall
building from here.
So what will be mine?
How will my eyesight make an angle?
Similarly, if I am looking at you in eye contact, then what
will be the angle of my eyes?
It will be zero.
Because I am looking at you in a straight line.
Right? So what will be the angle of the horizontal line
which will go through my eyes? It will be zero because it
is the same surface area.
Surface level is the same.
My eye level.
But if I am looking at something above my eye level, then I
will have to look up like this.
And then if I draw a line from my eyes, the thing which I
am looking at, let us suppose I am looking at a building or
a fan.
You look at the fan above.
So if you look at the fan, then you will look at it with
your eyes facing up.
So if you draw a straight line from your eyes to the fan,
then it will not be straight like it was on eye contact.
What will it be?
It will be drawn at an angle.
So when that line will be drawn at an angle, then will it
make an angle from the horizontal line below my eye level
or not? It will obviously make an angle.
This was our eye level.
Now we are looking above it.
So what will be this angle?
The angle which will form this angle.
If we look at the top, then the top side angle will be our
angle of elevation.
Elevation means to make an uphill or inclination.
And what will be the angle of depression?
In the same case, if you look below, then still you are
looking below your eye level.
So now if you are looking at the floor, then this
horizontal line will be drawn below.
Then what will be the angle between these two?
We will have an angle of depression.
I hope it is clear to you all.
What is the angle of elevation and angle of depression?
Whatever is our eye level,
I just gave you an example of eye level.
But we can also do this in any other thing.
Suppose if we are looking from below to above, we have a
surface level.
Any horizontal line will be our surface level.
In respect to this,
If we are looking at the top, then the top angle will be
our angle of elevation.
Because this is our horizontal line.
This is our surface level.
If we look at the bottom, this is our surface level.
But we are looking at the bottom.
This is our angle.
So this will be our angle of depression.
Give this one theta1 and give this one theta2.
So what will be theta1?
This is nothing but angle of elevation and this is angle of
depression.
If we see this as our eyes.
These are eyelashes.
This is eye ball.
If we see the top, this is our horizontal level.
Which will be straight.
This is that line.
If we see the top, this is the angle of elevation.
If we see the bottom, this will be our angle of depression.
Is the difference between the two clear? Top one is
elevation and bottom one is angle of depression.
So now we will do their questions.
Most of your questions on height and distance come on this.
And the tool to solve those questions is the one which we
have studied till now in trigonometry.
So let us do.
Let us see the question.
Your first question is this.
Read it once.
What is written? The angle of elevation of a ladder.
I had told you that it is not necessary that you have.
This is the example of ladder.
The angle of elevation of a ladder leaning against a wall
is 60 degree.
So we will make a ladder.
What we have to make?
First we have to make a wall.
The angle of elevation of a ladder leaning against a wall.
So we will make a wall.
Let us suppose this is our wall.
You must have understood.
What is the leaning against a wall?
It is a ladder.
So we will make a ladder.
Let us suppose this is our ladder.
Draw it properly.
There is no need to draw like this.
This is your ladder.
Against a wall is 60 degree.
Now how will we know what will be the 60 degree angle of
elevation? First of all, I had told you that if this wall
and this ladder are on the same level on the ground.
They are kept on the same floor.
So this is their same.
This is our eye of sight of eye.
The base level of the horizontal level.
So this is their floor.
So we have to find the angle of elevation from this.
I will draw dotted lines here.
Then what will be the angle of elevation? We will have this.
This is given to us.
60 degree.
Is the diagram clear?
How did you make it?
We have done till here.
And the foot of the ladder is 4.6 meter.
4.6 meter.
4.6 meter away from the wall.
Means, wall or foot of the ladder.
This is our foot of the ladder.
What is the distance between these two?
How far is it from each other? 4.6 meter.
So this distance is 4.6 meter.
4.6 meter.
Is it clear? 4.6 meter.
And then find the length of the ladder.
We have to find the length of the ladder.
That is this length.
So what is the wall? Wall is straight perpendicular.
There is no such wall.
It is straight.
So what is this? 90 degree.
90 degree.
Do you see a right angle triangle? This one.
Top of the ladder.
This is the whole ladder.
Its foot.
And the distance given is 4.6 meter.
And this is the surface of the wall.
If we see this, which I have shaded in black.
What do we see here?
We see a triangle.
And this 90 degree angle of the wall.
So what do we see here?
Right angle triangle.
Let's name it ABC.
And how much is theta given?
This angle C is 60 degree.
So if 60 degree, we are assuming it to be theta. We are
looking at it in respect to this.
So in respect to this, what is the base?
Base is BC.
That is 4.6 meter.
So what is the base for this? Base is for 60 degree.
4.6 meter.
What is the base?
BC.
For angle theta, that is 60 degree.
What do we have to find out? We have to find out AC.
To find AC.
What does AC look like for theta? Hypotenuse.
Hypotenuse is the length of AC.
So what do we have to find out?
We have to find out hypotenuse.
We are given the base.
And also given the angle.
So you see, which trigonometric ratio do we have for base
and hypotenuse?
We have to use trigonometric ratio because degree is also
given.
Angle is also given.
So hypotenuse and base, which formula do we use for these
two?
We use formula for cos.
What is the formula?
That is cos theta is equal to base upon hypotenuse.
This is the formula for cos.
Base is given.
What is theta?
Theta is given for 60 degree.
This is our given thing.
Theta and base.
If theta is given for 60 degree, then what will happen?
Cos 60 degree.
And what is the base?
Base is 4.6 meter.
That is BC.
And what is hypotenuse?
We have to find out AC.
So put the value of BC.
4.6 meter upon AC.
And what will be the value of cos 60 degree?
Look at the value of cos 60 degree in the table.
What will be the value of cos 60 degree? 1 by 2.
So value of cos 60 degree is 1 by 2.
Here I am solving.
1 by 2 is equal to 4.6 meter upon AC.
So what will be the value of AC?
AC value will be 2 into 4.6 meter.
That is 9.2 meters.
So what was AC? Length of the ladder.
AC was the length of the ladder.
So length of the ladder is 9.2 meter.
This is our answer.
Why it is in meter?
Because the given unit of this is also in meter.
So when we solve, what will be the answer?
It will be in meter.
So answer is 9.2 meter.
These were very easy questions.
Let us see the next question.
Next question is.
Read it once.
So the question is,
From the top of a house, 32 meter high,
Means we are talking about the height of the house.
Means there will be some height of the house.
And what is the height given to us? 32 meter.
The height of the house is 32 meter.
Means that house is 32 meter high.
If the angle of elevation of the top of a tower is 45
degree.
If we make a figure of any house.
Then we make a house.
This is our cute little house.
Ok.
What is the height of this house?
What is the height given to us? 32 meter high.
So this height is 32 meter.
This length is 32 meter.
So from the top of the house, Means from the top of the
house, Means if we see from here,
So what is the angle of elevation given to us? 45 degree.
When? When we are looking at a tower. So let us make a
tower.
Let us suppose this is our tower.
Ok.
If we see the angle of elevation from here.
So this is the horizontal point.
First the horizontal line.
Then.
Sorry.
This is.
Ok.
Then from here the angle of elevation is this.
If we see like this.
So what is the angle of elevation given to us? 45 degree.
Then next thing is.
And the angle of depression of the foot of the tower is 30
degree.
And when will the angle of depression be given?
When we look at the foot of the tower.
So how much is this?
This is what we have.
We will see here.
So this is our 30 degree angle of depression.
Then the height of the tower is.
What we have to find out?
Height of this tower.
This we have to find.
How will we do?
You see we have.
See this was the horizontal line.
of sight.
Line of sight means the horizontal line that I told you
about.
If you are making eye contact, the straight line will be
horizontal.
If we are looking at the example of the eye, we will call
it line of sight.
So this is your horizontal line.
This is line of sight.
This is also perpendicular.
This is also perpendicular to the tower.
So we have two right angled triangles.
If we have a tower, then we will call it C.
Then the line of sight that is being drawn here, if we look
at this tower straightly, then we will call it D.
So we have two right angled triangles.
And both the triangles are B, D, C.
We have to find out the length of AB.
The length of AB is 32 meters.
The height of AB is 32 meters.
The length of AB is 32 meters.
How did you understand?
Because the height of the house is parallel to the height
of the tower.
So the length of the tower is BD.
So the length of the tower is 32 meters.
If we get the length of AB, then we will sum both of them.
We will get the height of the tower.
We have to find out the height of the tower.
That is AD plus BD.
We have already given the length of BD as 32.
Now we have to find out AD.
To find out AD, we have to use both the angles of
depression and elevation.
So if you look at the triangle below, Triangle BD and C.
This is given as 30 degrees.
The length of BD is 32.
We have to find out the common length, CD.
The length of both the triangles is common.
So how do we find out CD?
The length of BD is 32.
What is the length of BD?
Is it perpendicular or base?
The base is CD.
So the length of BD is 32.
What is the length of BD?
It is perpendicular.
So what do we need?
Base CD.
So what is the trigonometric ratio in which it is
perpendicular and base?
What was the formula of tan theta?
Tan theta is equal to perpendicular upon base.
So what is theta for this triangle?
Which triangle are we talking about?
We are talking about triangle BDC.
In triangle BDC, the length of BD is 32.
We have theta which is equal to 30 degrees.
So what is tan theta?
It is perpendicular upon base.
So what is tan 30 degrees?
Tan 30 degrees is equal to,
What is perpendicular? Which we have to find out.
Which is given as BD.
And what is base?
It is CD.
Put value of BD as 32 meters.
And what is CD?
We have to find out.
So CD is as it is.
We will take CD there and tan 30 degrees will be taken down.
What will be the value of tan 30 degrees?
Sin 30 upon cos 30.
Either you remember or you can subtract it.
Sin 30 is 1 by 2.
And what is that?
What is cos 30 degrees?
What is root 3 by 2?
So what do we have? 1 by 2 divided by root 3 by 2.
That is 1 by root 3.
Tan 30 degrees is equal to 32 upon CD.
So what is the length of CD? CD will be 32 into root 3.
This root 3 will go up and CD will come here.
This much meters.
So length of CD is 32 into root 3 meters.
Length of CD is taken out.
Now I am erasing the above question for further calculation.
I hope you all have written this.
Write it once.
Okay.
The length which was common between both the triangles is
32 into root 3 meters.
Now what we have to find out? We have to find out AD.
So we will look at the right triangle above.
Right triangle ADC.
In right triangle ADC, we have theta.
We call it alpha.
Alpha is equal to 45 degrees.
We know the length of CD.
Now we got this.
We have to find out AD.
AD is alpha.
What is AD for 45 degrees?
AD is perpendicular to it.
CD is base for it.
For this, we have CD as base.
And AD as perpendicular.
So base upon perpendicular.
Or perpendicular upon base.
Which formula will be used?
Tan theta.
So tan alpha.
Because we have given name to 45 degrees.
So tan alpha will be perpendicular upon base.
So tan alpha tan 45 degree is equal to perpendicular.
What we have?
AD and base is CD.
Keep value of CD.
32 root 3 meters.
So what is AD?
Tan 45 degree is equal to 1.
Sin 45 upon Cos 45.
So 1 into 32 root 3 meters.
So we have AD's length.
32 root 3 meters.
Now AD's length is here.
If we add BD, tower's length will be here.
So height of the tower is AD plus BD.
That is 32 root 3 meters plus 32 meters.
If both the units are same, we can add.
So what is this? 32 root 3 plus 1 meter.
This is your height of the tower.
So this is your answer.
These two questions would have been easily solved.
Keep practicing these types of questions.
Till then, thank you for watching our video.
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