Trigonometric Identities: How to Derive / Remember Them - Introduction to Trigonometry Video Lecture | SSC CGL Tier 2 - Study Material, Online Tests, Previous Year

all right in this video I want to talk
about just some trigonometric identities
and at least how I remember them so you
know there's some things I think you
just kind of have to memorize you know
the basic ones sine cosine tangent sine
over cosine cosecant is 1 over sine the
secant is 1 over cosine and cotangents
cosine over sine I can't remember for
like cosecant things are a little
backwards I feel like you know the the
cosecant and the cosine should go
together so in my head cosecant it would
be nice if cosecant was 1 over cosine
but it's not it's kind of backwards so
the cofunction goes with just sine and
then it's kind of the opposite secant
goes with the the cosine so I think
these people get a little backwards
sometimes but you know whatever okay I
don't want to talk about these I want to
talk about some other ones as well kind
of the more you know the ones you have
to use to solve equations things like
that so kind of the most basic one I
think that if people tend to remember
one they remember sine squared plus
cosine squared equals one and the idea
is again the equation for a circle of
radius one the good old unit circle
remember that's just x squared plus y
squared equals one well for some angle
theta that point on the circle we refer
to that as cosine theta comma sine theta
well again this point is sitting on the
circle okay so that means that the
x-coordinate is gonna be given by the
value of cosine theta and the y
coordinate is given by the value of sine
theta so all we do is we're just using
the equation of our circle x squared
plus y squared equals one and we're just
plugging in cosine 4x + sine for y so
we'll get well cosine of theta squared
plus sine of theta squared equals one
and that's one of our very common
identities and again typically people
will write that as cosine squared theta
plus sine squared theta equals one okay
so there's one of my you
kind of first fundamental identities
it's easy to get a few others from this
no problem if we take this identity
cosine squared plus sine squared equals
one if we divide everything say by let's
divide everything by cosine squared so
I'm gonna take cosine squared theta and
divide it by well cosine squared theta
I've got sine squared theta and I'm also
gonna divide that by cosine squared
theta and then we've got 1 over cosine
squared theta so I'm just taking my
equation my first identity here and I'm
just dividing everything by cosine
squared theta okay if we do that well
cosine squared over cosine squared is
just 1 anything divided by itself this
is really sine theta over cosine theta
squared well that's gonna give us our
tangent squared theta against sine over
cosine is tangent but again it's being
squared likewise we could write 1 over
cosine squared is just 1 over cosine all
being squared well 1 over cosine again
is just gonna give a secant so in this
case it's gonna be squared so we get yet
another identity we get 1 plus tangent
squared theta equals secant squared
theta we can do the same thing we can
take our cosine squared plus sine
squared we can take that identity and
instead of dividing everything by sine
or by cosine squared let's divide
everything by sine squared theta so
we'll divide by sine squared theta we'll
divide by sine squared theta and we'll
divide by sine squared theta well again
cosine over sine that's cotangent but
again it's being squared so we'll get
cotangent squared theta sine squared
over sine squared is just 1 and then 1
over sine that's just a cosecant of
theta but again it's being squared so
we'll get cosecant squared so there's
another identity cotangent squared theta
plus 1 equals
secant squared theta okay so again just
by kind of knowing one I'm able to get
to others pretty quickly all I'm doing
is just dividing so you know nothing
worse than that let's look at some other
common identities here okay
so there's an identity that says that
sine of negative theta equals negative
sine theta so where does that come from
and likewise there's an identity that
says that cosine of negative theta
equals well cosine of theta and you may
remember this from symmetry of graphs
you know even and odd functions remember
if a function is symmetric about the
y-axis so I'm just drawing some random
graphs here if we have a function that's
symmetric about the y-axis remember we
say that's an even function and the
definition of an even function we say f
of negative x equals f of X and again
that makes sense because all that says
is if you go over you know if you go
over to some x coordinate f of x is the
y-value but if you go the other way well
negative x you're gonna get the same
exact y-value so that's all this
condition says geometrically remember if
a function is odd we say it's symmetric
about the origin
so an example of a function that's
symmetric about the origin
you know may be something like X cubed
and the definition there is we say f of
negative x equals well negative f of X
and again you can give just a little
geometric argument as to why those are
equivalent okay so those are the
definitions of even and odd symmetric
about the y-axis is even symmetric about
the origin
means it's odd well let's think about
the graphs of sine and cosine if you
think about the graph for example of
cosine remember cosine of 0 is 1 at PI
over 2 it's 0 at negative PI over 2 it's
0 and then we just get our normal little
wavy function okay so there's our graph
of cosine theta
well cosine of theta is an even function
it's symmetric about the y axis so that
means we can apply this definition and
all that says is okay so instead of f
well now we've got cosine that's our
function so we'll get this equivalent
definition it says cosine of negative x
is gonna equal well cosine of X so
equivalently cosine of negative theta
equals cosine of theta and again all
that is it's just a fact that you're
using you're just using the fact that
cosines even if you think about sine
theta so I mean a little Lussier putting
everything all over the place PI over 2
negative PI over 2 sine of 0 is 0 and
then you know it increases up to 1 and
it goes down to negative 1 on the left
so if you forgotten the graphs of sine
and cosine definitely want to look at
those well it turns out that sine of
theta is gonna be an odd function it's
symmetric about the origin
so equivalently you know if f is kind of
a generic function now we're using the
function sine so it says sine of
negative x would equal well negative
sine of X and again that's our other you
know our other identity up here so this
is just using the even the even
definition and here we're just using the
fact that sine is odd okay so that's
where those two formulas come from so
yeah you've got to remember something
else even an odd but I think sometimes
people just don't make this connection
ok so there's another nice little couple
of identities that we can make it use of
ok so again just ways to remember them
ok so we're thinking about the graphs
being even and odd let's look at a
couple others here you know a very
simple identity if you think about in
terms of the unit circle
okay suppose I'm at some angle here
theta
well again on the circle that's gonna be
the point cosine theta and sine theta
well if you add 2 pi to our angle theta
right if we add 2 pi to the angle theta
it's just gonna take us one full trip
around the circle and we're gonna be
sitting at the exact same location on
the circle so if we add 2 pi to our
angle well I claim that we're gonna get
the exact same value again you're just
sitting you're just sitting at the same
place on the circle so that's the idea
cosine of theta is gonna be the exact
same value as cosine of theta plus 2 pi
okay so that's gonna be another trig
identity and likewise sine of theta is
going to be sine of theta plus 2 pi and
again that makes sense all it says is
you're just adding 2 pi so you're back
at the same place where you started so a
couple of other a couple of other
identities here that I can remember
again just by thinking about the unit
circle
let's look at two more I guess
technically four more in this video
we're not gonna derive these I'm gonna
show you how I remember them but again
then I can show will show how we use
these to derive a bunch of others so
again here's two others that I just
simply I just know these okay and here
they are so it says sine of X plus y
equals sine X times cosine y plus cosine
X times sine Y so I have very few
mnemonics and my some people like
mnemonics to memorize things I never
really used too much here's one case
where I do use a mnemonic to remember
this identity I remember that sign is
something
notice something is kind of misspelled
their sign is something that switches
okay so why does that help me remember
well okay we're talking about sign you
know we're adding two things together
X plus y usually you know we learn
addition before subtraction so I think
you know X plus y that's kind of the
first thing I would do sign is so if it
starts positive sign is some thing some
you know referring to addition so what I
remember is if it starts positive the
identity stays positive so there's my
sum if it starts with the sum it stays
with a sum it stays addition and it
switches it switches I have sine and
cosine okay so I'm kind of switching
them out and then I switch it yet again
so sine gets the X cosine gets the Y but
then cosine gets the X and sine gets the
Y okay so everything's kind of switching
and the sum stays in there okay well I
remember likewise that if there's a
minus sign in there well you just keep
the minus sign as well so this one
little mnemonic helps me remember well
I've got two identities here now one
with the plus and one with the minus I
also remember that somehow cosine let's
look at cosine of X plus y I kind of
remember that that one is sort of like
the opposite okay and what do I mean by
the opposite well here if it started
with a plus it stayed with a plus here
we start with a plus and it's actually
gonna be a minus so in that sense it's
opposite notice to hear the sine and
cosine kind of play together right
there's a sign in a cosine a sine and a
cosine cosine doesn't do that it's gonna
stay we'll have cosine X cosine Y so
it's not switching it's just staying
cosine and then we'll get sine X sine Y
so again it's not switching okay so it's
in that sense that I think of it as
being the opposite
and again since we're doing cosine of X
plus y I just remember well it starts
with cosine
likewise if there's a minus sign you
know you have cosine of X minus y we'll
get an addition sign in there instead
okay so that's how I remember these four
identities okay that's my one little
terrible mnemonic sign is something that
switches and then I just again I
remember that cosine somehow does a lot
of the opposite stuff okay so I think
we'll stop here on this one I'm gonna do
another video again kind of showing in
fact well how can we use these
identities in fact to produce a whole
bunch of yet a whole bunch of other
identities