Sum and Difference of Two Angles in Trignometric Functions Video Lecture - Class 11

in this video lesson we are going to
learn the chicken very gracious of
Salman difference of the two angles
especially in this video we are going to
learn the cosine of the friends of two
angle right and that we are going to
derive like will derive adds cosign of
first angle times cosine of second angle
plus sine of first angle times sine of
second angle this we are going to derive
and with the help of this formula we can
easily find out the cosine of the sum of
two angles here a and B are two
different angle alright and that will
get this this is actually the result in
ER and we are going to derive these
results will get cos cost of a COS of B
minus sine of a times sine of B so how
this formula came let's take a look
right how to write this formulas take a
look for that I need to make a unit
circle let's say I have a circle this is
my rough circle hope it's looking good
really bad circle anyways this is my
here x axis positive x-axis positive X's
this is my origin all right and as I
said this is of unit length all right so
the radius of this you know this this
this is of one unit so one it means the
coordinate of this point let's say P
naught is 1 comma 0 all right
and now I'm going to imagine a point
here let's say this is 0.1 all right and
this point 1 I'm going to connect with
origin and let's say the point after
connecting this point 1 we got our line
o P 1 and that op1 radius is making the
angle of B with positive x-axis all
right and we need to find out the the
codit of this point P 1 so I'm going to
draw a perpendicular here and after
drawing perpendicular it is looking
almost like this you know exactly like
this with the angle B and here the this
is angle B so this opposite is the
perpendicular and this is the base and
this will be our hypotenuse the longest
side is always hypotenuse and if you
take a look properly then cosine of B
here is actually the base over the
hypotenuse right so the base over the
hypotenuse that is B over H and if you
look sine of B then that's actually the
perpendicular over the hypotenuse right
so pop any color over the hypotenuse a
little one thing on my friend here this
B is actually the base here but in this
figure B represents our X coded of the
point p1 all right
this we are going to find out so this is
actually our x coordinate and similarly
in the figure this is P but in this
figure if you look this is our Y color
of the point p1 right this is our y
coordinate so I already said that the
this circle is of hypotenuse one you
know the Cyprien is one saying or the
radius is actually of one unit we have
already considered that all right so
this this entire till here or till here
both our hypotenuse all right and both
have the one unit so simple V is going
to be at times cos B and s is going to
be sorry and P is going to be x times
sine B here this B is representing our x
coordinate which is now H is 1 so it is
simply cos B and this P is representing
Y so it is simply sign be alright it's B
so finally we got the coordinate of this
point p1 that's our cosine of B and sine
of B in the in the similar manner let's
say we have one more point here let's
say 0.2 all right and let's connect this
point 2 with origin right and let's say
the point 2 is making the angle of a
with the positive x-axis so so so that
what we can say the coordinate of this
point p2 is almost same in the same way
we are going to find out that that will
be our cosine of a and sine of a right
and now
again imagining are one more point here
right point three which is actually when
after connecting this with origin which
which is making angle exactly equal with
a difference B all right
this complete is angle a all right this
is complete angle B and this point three
is actually equal with a difference B so
the coordinate of this will be
definitely in the same way if you look
it will be cosine of a minus B and sine
of a minus B all right so we got the
three points point one making an angle
of B point two making an angle of a is
angle here bigger and B is a smaller one
and point three which is making angle a
difference B all right now here I want
to tell you one thing that the angles
angle angle p3o P naught is equal with
angle angle p2o P naught minus angle P 1
or P naught I'm going to shape it like
like this this I'm saying
this complete angle is this complete
angle is equal with angle B minus angle
is equal with angle a minus angle B all
right so and now this is equal with
angle a minus angle B all right so both
the angles are equal now this angle and
this both angles are equal so the kurds
subtended by these two angles will also
be equal right we had already learned
these things in geometry if the angle
like made by the two cards on the
central angles is equal then the kurds
are also you can I mean angles are equal
I'm central angles are equal then the
chords are equal and if chords are equal
then central angles are equal so here
angles are equal so we can say the kurds
the corresponding curves are also equal
so the
responding cards for the p3o peanut p3o
peanut is p3 peanut this card alright
this card alright let's say this is p3
peanut and the difference of this angle
is actually this one
alright this particular is this is the
angle you know this complete - this much
is this a - which this is this is also a
minus B so the representing card is this
p1 p2 alright this is p1 p2 so now we
can use the distance formula let's use
the distance formula to put the values
now p3 p naught
alright the length of this p3 period
card will be X 2 minus X 1 whole square
under root axially under root X 2 minus
X 1 y 2 minus y 1 we are using distance
formula that we had learnt that is X 2
minus X 1 all squared this is coordinate
and y 2 minus y 1 all squared right
inside the root so the same thing I'm
going to do here so our X 2 is cosine of
a minus V cosine of a minus V X 2 minus
X 1 is here 1 all right
whole square plus y 2 minus y 1 that is
sine a minus B sine a minus B minus 0
whole square so this is now equal with
p1 p2 right so p1 p2 we have yes so
again X 2 minus X 1 so that is cos a
minus cos B whole square plus again sine
a minus sine B whole square all right
sine a minus sine B whole square and
inside the root so now let's square both
side squaring both sides we get this
this root will remove up and we'll get
cos a minus B minus 1 whole square plus
sine square a minus B and that is going
to because a
- cause B whole square plus sine a minus
sine B whole square alright and now
let's use the formula of a minus b whole
square so it will be a square minus 2a b
plus b square and the sine square a
minus b and then here equals to we can
use same a minus b whole scale formula
so it will be a square minus 2a alright
minus 2 a B plus B Square and now let's
do the operation for this sine a minus
sine B again a square minus 2 a B plus B
square right this is sine square B all
right and now see here many things are
going to get a smaller value I mean this
cross square e minus V and sine square a
minus B will turn to one and this is
already one alright and now this minus 2
cos a minus B and equal width now cos
square a and this sine square a will
result to one because we know actually
this I'm using cos squared theta plus
sine squared theta equals to one right
I'm using this identity so cos square a
plus sine squared that's 1 and again
here cos square B plus sine squared B
that's going to be one all right
and now minus 2 cos a cos B minus 2 sine
a sine B all right so see here we can
cancel here something we can cancel this
one and one this plus one and this plus
one all right and further we can cancel
this 2 also but I'm frustrating minus 2
cos a minus B is equal with now he
here we can take - too common outside so
- - we'll come outside and it will be
cos a cos B and then plus sine a times
sine B all right so it's going to be
minus 2 n minus 2 will cancel up and
finally we'll get cos a minus B equals
to cause a times cos B plus sine a times
sine B all right so this is our required
identity or that the the formula that we
were about to find out you know and this
is how we find it up so cos a minus B is
cos a times cos B plus sine a times sine
B all right and now we can use this
formula to find out this cause a plus B
also how let's see here we have this
alright let's let's say here let's say
here let's say here cos a minus minus of
B all right now actually I change the
angle just I mean in the place of the
this B I put here minus B so it's it's
going to be now cosine of a times cosine
of minus B and then plus sine of a times
sine of minus B all right and now this
is cosine of a and cause now let me tell
you cause of negative angle is actually
cos positive angle you know cost cos
minus theta is equal to cos theta so cos
minus B will be simple Cosby and now
here this is minus sine B right and one
more thing here - sine theta is actually
minus of sine theta all right sine of
negative angle is minus sine of that
angle right so this minus B it is so it
will be minus and that - I'm putting
here and then sine a times sine B all
right and what is this cos a minus minus
of P this minus minus plus so it is a
plus B all right and this is our course
and this is how we need to find out cos
a plus B so this is the these are the
two result you know the
cosine of difference of two angles and
cosine of sum of triangles so this is
how we need to derive the formula I hope
you understand it and later videos we're
going to do the lot of applications of
these two formula and we're going to
find out the other formula also based
based on this and with the help of this
formula you know we're going to find a
lot of other trigonometric relations if
you can break identities all right so
see you in the next video goodbye