L3 : Inverse of Sin - Inverse Trigonometry Maths, Class 12 Video Lecture

hello friends this video on inverse
trigonometric functions part three is
brought to you by exam fear calm no more
fear on exam before watching this video
please make sure that you have watched
part 1 and part 2 so let's find the
inverse of sine function first the
national domain and range for sine
function is from R to minus 1 to 1 so if
you see the graph this is how the sign
looks like so you can take any real
number this if you see this graph this
is getting any real number the x axis it
can take any real number but the vise is
if you see the maximum value is 1 and
the minimum is minus 1 so this is minus
1 to 1 and this is a way now if I want
to restrict this restrict this to
interval such that I get all unique
values what I can do is if you take 0 to
PI range you see that for this and this
this in this point the output is same
it's correct because if you draw the
line God put the same so it won't do 101
so we can't take 0 to PI go ahead why we
can't exited why because if we take 0 to
PI what I call range as range as a
domain you see that two values maybe I
think here it is some value of things PI
back for something else PI by 4 and here
it is private you plus PI by 4 this
value both have same value of sine so if
you see this sign my wife 4 is nothing
but 1 by and then sine 3 PI by 4 you
have someone very good so move these
values PI by 4 and 3 PI over 4 lying
this range so we can't take this you
take this it won't be a 101 function so
let's try to take this one from PI by 2
3 by 9
so you take this you see all the ones
are unique so you can take these also
but the the one which we take generally
the national is minus Phi by 2 plus 5 it
this is the Nashville destruction see
you take this all zero it becomes 1 1
point if you take this also it because
one hour do I know all the values are
unique all the other unity for any for
any X you see you can't have 2 for a in
into X you can have you can have 2 same
way so minus PI by 2 to PI by 2 is also
1 valid domain plus PI by Del T plus 3
by 2 pi or it is also a valid though you
can see so there are multiple violent
domains unifying for which the function
can be inverse so the National one we
will take - why why do you do you must
permit take this the function becomes 1
over now you see if you restrict this to
minus PI by 2 plus PI by 2 this becomes
1 1 and not tools and range now we have
to restrict the range of ranges minus 1
to 1 so in this in this whole thing if
you restrict the range to minus 1 to 1
and if you restrict the domain to minus
PI by 2 T 2 s PI by 2 the whole thing
becomes invertible function because that
becomes 1 1 and on to if you take any
value in this domain from minus pie by 2
to plus PI by 2 it will give a unique
value only so if you see the restricted
domain and range for the sine function
is minus PI by 2 pi by 2 this is the
domain and the range is minus 1/2 so in
this case if there is a 101 and immune
function so it is invertible so if you
flip this if you inverse this you write
the sign reverse you have to just flip
this so this becomes the rain becomes
domain here and this becomes the domain
good range here so what we got for the
sine inverse function
aspectek domain and range is minus 1 to
1 this is my domain and range becomes
minus 500 plus so as I have told that if
sign is restricted in any of the
interval minus 3 PI Omega t minus Phi by
2 or pi by 2 G by I do all these will
give you one set of inverse function
because all this say you see are 101 and
andö but minus PI by 2 to PI by 2 is
called principal value branch
please note this is called principal
value and these are branch if you see
minus 3 PI by 2 will be somewhere here I
think yeah this one you - man is that
all these if you see these range are a
normal intervals the nominal doubles for
which the sign of value are gone on and
on but this particular one is called
principle valued right there are so many
branch for this the sine function is
invertible minus PI by 2 to PI by 2 is
called the principal value branch and
the other branch icon normal branch also
if Y is equal to sine inverse X then
sine Y is buddy it's Lisa
when if you see Y is going to sine
inverse X then by then sine Y is going
to ask the second nation should note
this you say Y is equal to sine Y or 6
take sine inverse here so this becomes
sine the sine Y become just let us
understand the graph of sine inverse of
G so we got the sine inverse so we can
form the sine inverse graph also so if
this is y is going to sine X you see
this becomes Y square design versus how
we just topple it so if you see this
here the y axis contains minus 1 to 1 in
x axis contains all this - and all this
minus PI by 2 plus PI by 2
mothers it is to upper lip here the YX
is contains all this values minus 5 by 2
pi minus 2 pi minus 2 by 2 pi fil and X
contains minus 1 1 just flip it that's
up and you get this graph and this graph
is nothing but my sine inverse x
function now if you see you add these
two so this guy's actually if you see is
obtained by actually finding the values
Y is equal to sine inverse X we have
plotted the values we got the values and
we have plotted it and we found this so
this was done by actually finding the
values of y is equal to sine inverse X
by putting different values minus PI by
2 pi and all we got this guy now after
plotting is glad we observe that this is
nothing but a mirror image of this graph
along y is equal to X line so let's see
this so graph the inverse function is
also obtained by getting the mirror
image of the original graph and gamma is
going to X this is my original graph y
is going to sign it you find the bias
but design project you just have to draw
this wise where this X line and just get
the middle so if you get the middle of
this line mirror of this measure of this
along this line what you will get is
this graph and this is nothing but Y is
equal to sine inverse so this is what
you observe that sine inverse graph is
nothing but whether image of sine graph
along line y is it X so that is all
about sine function what we also have
the sine function was not 1 1 on 2 or
the national domain because notion
domain and range
Marshall do mean whether all Riggleman
ring was -21 because not one of our not
to what we did was to make it invertible
it restricted the domain and range and
then we found that by resting thing then
the waves made the
some chip-in were to bill and then we do
the graph of y is equal to sine x and y
is equal to sine inverse X and then we
observe that the graph of y is equal to
sine plus X is nothing but the mirror
image of y is equal to sine sine X along
right
y is equal to X now let's do the same
thing for cos x thank you visit exam
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