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Hi!

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I'm Pritish

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Today, we will be talking about Trigonometry.

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This is the first video on Trigonometry.

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I would like to start with "Why do we want
to study Trigonometry?"

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and "What is Trigonometry?"

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You would certainly know that Euclidean geometry
is very fundamental to how we think about

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nature.

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The way we visualize things in Physics, for
example, we use Euclidean geometry in the

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way we think.

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And often, Euclidean geometry requires some
amount of innovative thinking to reason about

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problems.

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And often when you want to talk about Physics,
you don't want geometry to be

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your bottleneck in thinking about Physics.

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So Trigonometry and Coordinate Geometry are
two subjects which try to make a more systematic

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study of Geometry.

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And what is "Trigonometry"?

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Trigonometry is about a study of angles
in an "algebraic" way.

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You would feel more comfortable doing Algebra

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Algebra is something you can do in a more
mechanical way.

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Whereas, Geometry often requires an innovative
leap.

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Using trigonometry, you'll be able to reason
about geometric problems in a more algebraic

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way.

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Let's start with the very basics of what is
an angle.

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You know from basic Euclidean geometry, that
if you have a circle of radius 'r',

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then the perimeter of the circle is equal
to 2.\pi.r.

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We know this from basic geometry.

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A basic reasoning will tell you that
length of the arc of half a circle

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is \pi.r.

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Similarly, if I take a quarter circle, the
length of the arc is (\pi/2).r.

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In general if I pick out an angle of, say,
\alpha degrees (like this could be 60 degrees),

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then you can easily reason that the length
of an arc with angle \alpha degrees is (\alpha/360).(2.\pi.r).

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This is just a basic ratio and proportion
argument.

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Let me write this as ((2.\pi.\alpha)/360).r.

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We are more generally taught since school
to think in terms of degrees.

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The full circle has 360 degree.

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But now when we have to write the length of
an arc,

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we have to write an annoying expression ((2.\pi.\alpha)/360).r.

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It would be much nicer to call this expression
((2.\pi.\alpha)/360) as the "angle" itself.

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Instead of \alpha degrees if we could call
((2.\pi.\alpha)/360) as the "angle", then

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it would be much nicer.

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As there are different ways to measure objects,
for example, length, you can measure it in

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meters or inches.

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similarly this is a different unit for measuring
angles.

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This is angle in "radians".

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Thinking in terms of radians is very important.

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My math teacher used to tell me that when
you are learning a new language,

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for example, if I am trying to speak in English,

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if I want to learn it well, I should not think
in a different language.

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For example, I should not think in Hindi and
translate to English every time I am speaking.

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You should really start thinking in English,
if you want to learn English well.

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Similarly here, thinking in terms of radians
is often the better thing to do

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because now the length of arc with angle \theta
radians is simply (r.\theta).

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So that's why thinking in terms of radians
is often the more natural way to think about

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angles.

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Just to be more familiar with this conversion,

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lets see that 360 degrees is '2 \pi' radians.

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The perimeter of the entire circle is '2.\pi.r'
and '2\pi' is the angle of the entire circle.

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Similarly, 180 degrees is \pi. 90 degrees
is \pi/2 and so on.

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So we just saw how to measure angles in degrees
and radians

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and we said that radians is the more natural
unit to think about angles.

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So that now the length of the arc with angle
of \theta radians is just r.\theta.

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In the rest of the video, we will talk about
trigonometric ratios,

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we'll define what they are and we'll talk
about some trigonometric identities.

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These are the algebraic tools which will help
us reason about angles in a more algebraic

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way.

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Let's look at an angle \theta.

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Whenever we write \theta as an angle, it is
assumed that it is in radians,

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as we know that radians is the more natural
unit to think about angles.

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Let's say the radius is 'r',

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and we just saw that the length of the arc
is r.\theta.

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Now there are other interesting parameters
that you would want to know about the angle.

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For example, if we drop a perpendicular from
the top vertex,

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what is the height of this line segment?

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Let's call the vertices as A, B and C.

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This side BC is "opposite" to the angle.

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The side AC is the side that is "adjacent"
to the angle.

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And the side AB is the "hypotenuse".

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So we would like to know the height of the
line segment BC is.

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One thing that you know from similarity of
triangles is that

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suppose I have two triangles like this.

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You know from similarity of triangles that
BC/AB = DE/AD.

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(Note that angles BCA and DEA are right angles.)

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These are two triangles which have all three
angles the same,

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and therefore they are "similar" and hence
the ratio of sides are equal.

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Because of similarity of triangles, this ratio
is purely a property of the angle,

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and is not related to any of the sides.

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This ratio is defined as the "sine" of the
angle.

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To remember it more easily, let's call sin(\theta)
as opposite/hypotenuse.

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Now we can see that since the hypotenuse is
'r', the length of the opposite side is now

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'r.(sin \theta)'.

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Similarly, now we can define the ratio of
the adjacent/hypotenuse.

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So that is defined as cos \theta.

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Again, since the hypotenuse is 'r', the adjacent
side is of length r.(cos \theta).

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In terms of this ratio, this will be AC/AB
= AE/AD.

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And
by similarity of triangles, this is just a

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property of the angle,

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and does not depend on the sides of the triangle.

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Now, we'll see our first trigonometric identity.

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From Pythagoras' theorem, we know that for
any right angled triangle,

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the sum of squares of the two sides is equal
to the square of the hypotenuse.

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So let's write down here : (opp)^2 + (adj)^2
= (hyp)^2.

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This is what Pythagoras' theorem tells us.

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If we divide by (hyp)^2 throughout, we get
that (opp/hyp)^2 + (adj/hyp)^2 = 1.

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Now, if you look at our definitions of sin
\theta and cos \theta,

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this tells us that, (sin \theta)^2 + (cos
\theta)^2 = 1.

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A usual convention to write (sin \theta)^2
is that

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you don't write the brackets all the time.

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So what we do is, write it as sin^2(\theta)
and cos^2(\theta).

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So as to not confuse with, for example, sin(\theta^2).

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This would mean sin(\theta^2), whereas this
is (sin \theta)^2