Hello everyone, welcome back to a new video.
In today's video, we are going to discuss the topic of
Height and Distance.
Before discussing the topic of Height and Distance, what we
should know in Maths is Trigonometry.
So, before discussing those questions, let us first learn
the basics of Trigonometry.
Let us learn all its formulas, which we have learnt in 10th
and 11th class.
Let us do a little bit of that.
Then, we will discuss the questions related to Height and
Distance.
So, what is Trigonometry?
You must have heard of Trigonometry.
If you have not studied it, then you must have heard of it.
Trigonometry, in which we have sine, cos, tan, theta.
You must have heard of all these.
So, Trigonometry is a branch of Maths, in which we study
about what?
We study about the measurement of angles.
Sorry, we study about the measurement of triangles.
In Trigonometry, there are three angles.
So, triangles have three angles.
Which figure has three angles?
Triangle has three angles.
Now, which triangle will we study in this?
You must have studied about different types of triangles.
One is Equilateral Triangle, in which all sides are equal.
Scalene Triangle, in which there is no scalene triangle.
Isosceles Triangle, in which two sides are equal.
And, one you must have studied, right angle triangle.
Right angle triangle, we will mostly use in Trigonometry.
So, what is the property of right angle triangle?
Which triangle is right angle triangle?
In which, one angle of triangle should be 90 degrees.
If one angle of any triangle is 90 degrees, then it will be
called right angle triangle.
If one angle of triangle is 90 degrees, then the other two
angles will be less than 90 degrees.
Why is it so?
Because, you must have heard about the sum of all the
angles of any triangle.
It is not necessary to use only right angle triangle.
In any triangle, the sum of all the angles is 180 degrees.
It is not less than 180 degrees, nor more than 180 degrees.
It is equal to 180 degrees.
So, if one angle of triangle is 90 degrees, then what
should be the sum of other two angles?
The sum of other two angles should be 90 degrees.
So, 90 will not be equal to 0.
Otherwise, it will not remain a triangle.
So, what should be the sum of other two angles?
The sum of other two angles should be less than 90 degrees.
So, what should be the sum of other two angles?
The sum of other two angles should be 90 degrees.
So, 90 plus 90 will be equal to 180 degrees.
It is the sum of all the angles of triangle.
So, this is the first property of right angle triangle.
Now, we will do the trigonometry in right angle triangle.
You should know what is right angle triangle.
Now, I have drawn a right angle triangle.
What else can you learn from this? Let me draw it again.
One line is here.
What is this? This is right angle triangle.
It is different from the basic triangle.
I have inverted this side.
Now, what is the 90 degree angle?
This is 90 degree angle.
This is 90 degree.
Both the lines are 90 degree perpendicular.
What are the other two angles?
One angle is here.
The other two angles are here.
Let us give a name to this triangle.
Names are more important than things.
So, we will give a name to these vertices.
A, B and C.
Give a name to this.
A, B and C.
So, what do we have?
This is right angle triangle.
What is the name of this triangle?
A, B and C.
What is the name of this triangle?
A, B and C.
We keep the 90 degree angle in the middle.
So, this is right angle triangle.
A, B and C.
We have other two angles.
Angle C and angle A.
Let us give a name to angle C.
Theta.
What is the representation of this?
This is called theta.
This is called angle alpha.
We have two angles.
Angle C and angle A.
We have given their names.
Theta and alpha.
We know the value of angle B.
That is 90 degree.
So, we know its value.
So, this is right triangle.
A, B and C.
Now, let us see the sides.
What is the name of these sides?
The longest side is called hypotenuse.
The rest of the sides are called base and perpendicular.
Which side is called base and which side is called
perpendicular? How will you know that?
You will know that from the angle.
For which angle are you talking? Then, according to that,
we will find out the base and perpendicular.
Let us suppose we are talking about this angle.
Angle theta.
We are talking about the angle on C.
We are talking about angle theta.
So, which two vertices are touching angle theta?
One is touching BC line.
And the other is touching AC side.
We have already told the name of AC.
The longest side is hypotenuse.
So, where is the rest of the side supporting?
The rest of the side is supporting BC.
AC and BC.
So, AC is hypotenuse.
What is BC? The side on which BC is supporting is called
the base.
So, BC is the base for angle theta and angle C.
Similarly, the rest of the side is supporting BC and AC.
So, the side on which BC is supporting is called the
perpendicular.
Perpendicular for theta.
Why perpendicular?
Because if we are looking at it from here, it is making 90
degrees.
Two lines are making 90 degrees.
One is BC and the other is AB.
So, what is perpendicular for theta?
For theta, we have perpendicular side AB.
AB is perpendicular for theta.
And AC is hypotenuse.
So, this is the name of the three sides of the three lines.
For whom?
For theta.
Now, if I ask you the same thing, for angle alpha, what
will be the base and perpendicular for angle alpha?
Everyone knows hypotenuse.
For alpha, it will be the same.
Now, for this, what do you have to tell?
Base and perpendicular.
Take 10 seconds and tell me quickly.
What will be the base and perpendicular for alpha?
So, I will write here for alpha.
I hope it is visible.
It is visible.
So, for this, tell me the base and perpendicular.
I hope everyone has done it.
Verify it once from here.
This is angle alpha.
On which two sides has it taken support?
AC and AB.
AC is our hypotenuse.
So, what do we have?
What will be the base for this?
On which side is the other taking support?
On AB.
So, for this, we have the base side AB.
And what will be the perpendicular?
What do you see opposite to this?
This side, that is BC.
So, for BC, what will be the perpendicular?
And hypotenuse will be the same, AC.
So, I hope you have understood. the of the whole square is
Pythagoras Theorem.
I hope it's clear.
Next moving on to the main thing in our trigonometry that
is trigonometric ratios.
What are trigonometric ratios?
First of all, you know that.
Trigonometric ratios are the ratio of sides.
If we take the ratio of sides, then what will it give us?
It will give us a ratio, if we write divide in fraction or
give a fraction, then we will see that fraction.
We will see whose fraction will be equal to which, it will
be equal to a particular thing.
We will call that trigonometric ratios.
We will see whose equal to sin theta, cos theta or tan
theta.
We will see that.
So next moving on to trigonometric ratios.
How many ratios do we have in total?
We will have 6 ratios, trigonometric ratios.
So first of all, what will you make?
Make a triangle, right angle triangle.
This is our triangle.
A, B and C.
This is our hypotenuse.
Now we will do it for a particular angle.
Let us suppose we are doing it for angle C.
We will name the angle C as theta.
Angle theta.
So in which respect do we take out trigonometric ratios?
We write in respect to some angle.
What will we write with sin?
Sin theta or sin x or sin c.
Whatever is the name of the angle, we will write
trigonometric ratio with it.
So we wrote theta.
So what is the base for theta? BC.
So this would be our base.
And this would be our perpendicular.
This is our perpendicular.
Will all this be clear?
Whatever I have written till now, is related to what we
read last. So this is our right angle triangle ABC.
And theta is that.
In respect to theta, we have written base and perpendicular.
Now the ratios we have are 6 ratios.
I will write here what all are there.
First of all, sin theta.
Sin.
And for theta, we are taking out theta.
If we are taking out for this one, for angle A, then
whatever is the name of angle A, or write sin A.
Because now only one angle is being made on A.
So we can write.
Otherwise, if it had another name, alpha, then we would
have written sin alpha for that.
Now we are doing for theta.
We are not writing alpha.
So sin theta.
Second we have cos theta.
Then we have tan theta.
These are the main three.
With the help of these, we get the remaining three also.
I will write that also now.
First, let us write what is it.
So what is the formula for sin theta?
We have the formula for sin theta.
Perpendicular upon base.
Sorry. Perpendicular upon hypotenuse.
Hypotenuse.
So that is.
What will we get for sin theta? What is perpendicular for
theta?
Perpendicular is AB.
So this would be AB.
Divided by hypotenuse AC.
So this is our sin theta.
Similarly, there is formula for cos theta.
In this, perpendicular upon hypotenuse is
What will be the value of cos theta?
Other side. That is base upon hypotenuse.
So this is our cos theta.
So if we see from here, what is the value of cos theta?
Base is BC for theta.
And hypotenuse is AC.
So BC upon AC is equal to our cos theta.
Then tan theta.
What is tan theta?
So divide these two.
So what will happen?
If we divide these two, hypotenuse and hypotenuse will
cancel out.
What will be the answer we will get? Perpendicular upon
base.
What is perpendicular for theta?
AB divided by BC.
So this is our formula for tan theta.
We have written for tan theta.
I will also tell you a short trick to remember.
How can you remember?
Now you write this formula.
Okay? The remaining three formulas.
I will up the figure and write it in front of this.
The remaining three trigonometric ratios that we get.
That comes.
This is sin theta.
One comes in reverse.
That is cos theta.
Then one comes in sec theta.
Then one comes in cot theta.
What will be sec theta?
Reciprocal.
What is reciprocal? One upon that thing becomes reciprocal.
So what will be the reciprocal of sin theta?
One upon sin theta.
So cosec theta is nothing but one upon sin theta.
That is what I told you that these three are main.
Or you can put only two.
Sin and cos are main.
You can remember everything about them.
You can remember the formula.
With their help, we can find out the remaining six.
Okay? So one upon sin theta.
So reciprocal it.
Hypertenuse upon perpendicular.
So what is this?
Hypertenuse upon perpendicular.
Or ac upon bc.
You can write ac upon bc.
Sec theta.
Similarly, what will be sec theta?
Cos is reciprocal.
So one upon cos theta is our sec theta.
So what is this?
Hypertenuse upon base.
Then cot theta.
You can write cot theta as cos upon sin.
Or one upon tan.
We have reciprocal of tan.
And what will be the reciprocal of tan?
Tan is sin upon cos.
So this becomes cos upon sin.
So what is its value? Base upon perpendicular.
Cot theta is base upon perpendicular.
Everyone should remember these formulas in their tips.
Whose formula is it?
Sin and cos.
And you should know their relation.
Who is whose reciprocal?
Then it will be easy.
So if we see its short trick.
So what can we see in its short trick?
What do you have?
Sin.
Sin theta.
Cos theta.
Tan theta.
You have these three.
If you know these three, then you will be able to solve the
remaining three also.
Then what we will do is.
Pandit.
Badri.
Prasad.
Har. You just have to write their initials, Pandit
Badriprasad Har Har Bhole.
And below this, what should you do?
Draw a line.
Because it is in ratio, fraction, divide.
So what is this?
P by H is a formula for sin theta, perpendicular upon
hypotenuse, for cos, base by hypotenuse and for tan,
perpendicular upon base.
When you get to know about them, you can reciprocate them.
You will get cos theta, sec and cot theta.
So now you know all this.
Now let us move ahead to their values.
We should know the values of some main angles.
That is, what will be those main angles?
Those main angles will be 0 degree.
Now the angle, you can measure it in distance.
You can measure distance in kilometer or meter.
Similarly, if you talk about the measurement of angle, then
you can measure any angle in which unit?
You can measure angle in which unit?
What do you call it? You call it degree?
Do you measure in degree?
Like I was saying 90 degree, so what you have studied till
now, you must have studied degree only.
Now you will study a new thing, you will know one more
thing, that will be your radian measure.
Tell degree in radian.
Radian measure.
So in degree, everyone will know that it will be 0 degree.
Your main degree will be 0 degree.
One will be 30 degree, on which you will know the values.
0 degree, 30 degree, 45 degree, 60 degree and 90 degree.
On these, you will have to memorize the values of sin theta.
If you do sin theta, the rest will automatically come out.
Now what is radian measure?
I will tell you the conversion of radian measure.
Like there is a relation, 1 km is equal to 1000 meters.
And then whatever is given, we convert it.
Like if we are given 100 km, we can change it in meters.
With the help of this relation.
Similarly, what will we do in degree and radian measure?
We will know a relation.
So what is the relation?
Our 180 degree is equal to Pi.
Everyone must have heard of Pi.
To find the circumference of a circle.
2 Pi r. You must have heard of Pi.
So our 180 degree is equal to Pi radian.
Pi radian.
We denote radian with a small c.
So 180 degree is Pi radian.
If 180 degree is Pi radian, then what will be 1 degree?
What will be 1 degree?
The same unitary method.
Pi radian divided by 180.
Then we will get the value of 1 degree.
Now when we get the value of 1 degree.
Then we can find the value of any of these in terms of
radian.
In terms of radian.
So we find that.
What happened to you?
This is our degree measure.
Where did it go?
This is our degree measure.
This is in degree.
If I want to write in radian.
Then 0 degree.
If 1 degree is Pi divided by 180.
Pi radian divided by 180.
1 degree.
So what will be 0 degree?
In 0 degree, nothing will matter.
0 will remain in radian.
So 0 degree will be equal to 0 radian.
30 degree will be equal to.
If 1 radian is equal to 30 degree.
30 degree is equal to.
How much will it be?
Multiply by 1.
30 into Pi by 180.
This gets cancelled out.
1 becomes 6.
So we have 30 degree in radian.
Pi by 6.
Similarly, 45 degree.
If I remember, I will write.
Pi by 4.
As I have taken out 30 degree.
Multiply 1 degree by 45 degree.
So Pi by 4.
60 degree is Pi by 3.
90 degree is Pi by 2.
How can you take this out?
If you want to take it out quickly.
Instead of Pi, keep 180 degree.
180 degree divided by 6.
30 degree.
180 degree divided by 4.
How much? 45 degree.
Similarly, so on.
So this is taken out.
In terms of radian.
Degree in terms of radian.
Now we will make a table.
Whose? Sine.
Cos. Tan.
Rest you can take out with their help.
I will make you write these three.
So this is your E2.
All of you must have written this.
Write this conversion quickly.
First of all, we write whose? Sine.
Sine.
Any angle?
Theta.
Sine theta.
Then we will do cos theta.
Then we will do tan theta.
How will you take out sine theta?
First of all, you write all the numbers.
What to write?
What to write? 0, 1, 2, 3, 4.
Have you written? 0, 1, 2, 3, 4.
Now what to do?
Divide all these by 4.
We will divide 0.
No effect will be there.
1 divided by 4.
2 divided by 4.
3 divided by 4.
And 4 divided by 4.
Now what to put outside these?
Under root.
What to put outside these?
Under root.
Now see, what has become? 0 as it is, 1 by under root 4.
Root 1 is equal to 1.
Root 4 will be equal to 2.
So, this becomes 1 by 2.
Then, what has become?
Root 2.
What will become? 1 by root 2.
Because 2 will cancel.
Only 2 will remain in denominator.
So, this becomes 1 by root 2.
What has become?
Root 3 by 4.
Root 3 by root 4.
That is 2.
Root 3 by 2.
And what has become? 1 has been cancelled out.
So, this has become under root of 1.
That is 1.
So, this has become your sign table.
That is 0, 1 by 2, 1 by root 2, root 3 by 2 and 1.
Remember this in this sequence.
Now we have written the sign.
You don't have to do anything about the cause.
You have to write in reverse way.
You have started from 0 and went till 1.
In this, you have started 1 from behind.
Like this.
So, what will happen?
First, you will write 1.
Then, you will write root 3 by 2.
Then, you will write 1 by root 2.
Then, 1 by 2.
And then, 0.
So, this has become your sign table.
That is, cause 0 degree is 1.
Cause 30 degree is root 3 by 2.
Cause 45 degree is 1 by root 2.
Cause 60 degree is 1 by 2.
And cause 90 degree is 0.
Now, tan.
What was the formula of tan?
Sin upon cause.
So, divide both the values.
0 upon 1.
0.
1 by 2 upon root 3 by 2.
2, 2 cancel.
So, 1 by root 3.
1 by root 2 divided by 1 by root 2 cancel out.
Cause 1.
Root 3 by 2 divided by 1 by 2.
That is, root 3 by 1.
Which is equal to root 3 only.
And 1 divided by 0.
That is, 1 divided by 0.
That is, what will happen?
Have you ever read 1 divided by 0?
Till now, whatever fraction we used to read.
We used to take the same definition.
What should be your denominator?
Denominator should be non-zero.
But, here it is 0.
So, what will happen?
This value does not exist in our mind.
Or in our notes.
So, what will we write for this?
Does not.
Exist.
Does not exist.
Either you can denote it with any finite number.
If we have divided by 0.
If you want to denote this.
You can denote this with infinity sign.
Either you write does not exist.
Or you write infinity.
It is the same thing.
Now that we know the sign, cos, and tan, what will you make?
Now it is your homework to make the table of cosec, sec,
and cot.
How will you make that? With the help of their formula.
That is, what will become of cosec? 1 by sine, so
reciprocate it.
Sec's 1 by cos, cos, reciprocate it.
Tan, cot's 1 by tan, so reciprocate the line of tan.
So, that's all for today's video.
You remember this much, then we will move ahead in the next
class and ask questions.
Thank you for watching the video.
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