Trig identities part 2 (parr 4 if you watch the proofs)

Trig identities part 2 (part 4 if you watch the proofs)

Welcome back. I'm now going to do a bit of a review of everything we've learned so far about maybe even trigonometry and trig identities
and then we'll see if we can come up with a... maybe use what we already know to come up with a a couple more trig identities
so you know we know that from soh-cah-toa we know that
sine of theta is equal to the opposite over the hypotenuse, right like if we were to... let me draw a triangle here
if I were to draw a triangle here, whoops, oh come on, there you go...
look at that
OK so this is theta,
this is the opposite
this is the adjacent
this is the hypotenuse
right?
then sine of theta is equal to the opposite over the hypotenuse
cosine of theta, this is basic review, hopefully at this point
is the adjacent over the hypotenuse
the tangent of theta is equal to the opposite over the adjacent, which is also equal to the sine of theta over the cosine of theta
and we showed this a couple of videos ago
and then these are kind of almost definitional but
the cosecant of theta is equal to the hypotenuse over the opposite
which is the same thing as one over the sine of theta
you can just memorize this
I mean, I find it silly that there is such a thing as cosec, I guess it is just for convenience because everyone knows it is just one over sine of theta
the same thing for secant.
secant theta. It's really for convenience... instead of having to say you know
in the case of secant "oh that's one.. " you know if you end up with the equation one over cosine of theta you can just say "oh that's the secant of theta"
I think it actually has some obvious properties and if you were to draw a unit circle and all that too
but anyway, so that's equal to the hypotenuse over the adjacent
which is equal to one over cosine of theta
and then of course, cotangent of theta is equal to the adjacent over the opposite which is equal to one over tan theta
and of course that's also equal to cosine of theta over sine theta, right
it's just the opposite of the tangent of theta, right?
or... that's the same thing as... what
that's the same thing as the secant... no no no
it's the same thing as the cosecant of... no... let me make sure I get this right
it's the same thing as... I just want to get the inverses...
well let's prove what it is... actually... I always confuse myself...
so this is the same thing as one over the secant of theta
over one over the cosecant of theta, right?
secant of theta, cosecant of theta... and then that equals
the cosecant of theta over the secant of theta
I wouldn't waste your time memorizing...
so we know that the cotangent of theta is equal to one over tangent of theta
is equal to the cosine over the sine
and it also equals the cosecant over the secant
and I wouldn't worry about really memorizing this. you could derive it if you had to, as you can tell, I really didn't have this memorized either
and we also learned in previous videos that the sine squared of theta plus the cosine squared of theta is equal to one
that just comes from the Pythagorean theorem
and then if you play around with this a little bit you also get that
the tangent squared theta plus one is equal to the secant squared of theta.. you can actually go from here to here if you divide both sides by cosine squared
so we know that
and then if you've watched the last two proof videos I made we also know that
the sine of 'a' plus 'b' is equal to the sine of 'a' times the cosine of 'b' plus....
let me erase some of this because I don't think that is an important trig identity. you can derive it on your own
I just wanted to show you that you could figure it out
I'm using too much space
OK. now I have space
let me find that blue color I was using and make sure my pen is small
OK that is the sine of 'a' times the cosine of 'b' plus the sine of 'b' times the cosine of 'a'
and you might want to just memorize that
this actually becomes really useful when you start doing calculus because you have to solve derivatives and integrals... and you might happen to know the identity
and it's not that hard to memorize, right?
it's the sine of one of them times the cosine of one of them plus the other way round.
that's all this is
we also learned that the cosine of 'a' plus 'b' is the cosine of both of them minus the sine of both of them.
so that is equal to cosine of 'a' times the cosine of 'b'...
and I proved this in another video. hopefully to your satisfaction
...minus the sine of 'a' times the sine of 'b'
these are pretty useful because from these we can come up with a bunch of other trig identities
for example
what is the sine of two 'a' ?
the sine of two 'a'. well that's just the same thing as the sine of 'a' plus 'a'
and if we use trig identity up here that is equal to
sine of 'a' cosine of 'a' plus the sine of 'a' cosine of 'a' right?
I just used this sine of 'a' plus 'b' identity up here and I just... well 'a' and 'b' are both 'a'
and what does this equal?
well this is two terms which are both sine of 'a' cosine of 'a'
so that just equals two sine of 'a' cosine of 'a'
so we now have derived another trigonometric identity that might be in the inside cover of your trig or actually your calculus book
let me... all of these actually... I could draw a square around all of them
well let's do another one
let's figure out
it's actually... once you have a bit of a library of trig identities you can really just keep playing around and seeing what else you can..
and I encourage you to do so
and you can be amazed how many other trig identities you can come up with
for example let's do cosine of two 'a'
cosine of two 'a' is equal to cosine of 'a' plus 'a', right?
and cosine of 'a' plus 'a'... what did we say... it's the cosine of both of the terms times each other minus the sine of both of the terms
so that equals cosine of 'a' cosine of 'a', right
minus sine of 'a' sine of 'a'
we saw... this identity was the cosine of 'a' plus 'b' identity
minus sine of 'a'
so what is this?
this is equal to cosine squared 'a' minus sine squared 'a'
now that's interesting... we could play around... this is interesting this is the form 'a' squared minus 'b' squared,
right?
so that's also the same thing as ( 'a' plus 'b' ) times ( 'a' minus 'b' )
so that's the same thing as cosine of 'a' plus sine of 'a' times cosine of 'a' minus sine of 'a'
I don't know this isn't really a trig identity... I'm just showing you can play with things
the cosine of two 'a' is equal to cosine of 'a' plus sine of 'a' times the cosine of 'a' minus sine of 'a'
so the sum of the cosine and sine of 'a' times the difference
that's just interesting.
I'm showing you that what's fun about trigonometry is that you can kind of keep playing around with it and actually that's probably...
that is how all of the trig identities were discovered
so let's say that we have... you know we want to figure out what cosine of let's say negative 'a' is
well let me draw a right triangle
oops... that's almost a right triangle
and let's say this angle is 'a', right?
so negative 'a' if we use a unit circle it looks something like this, right?
negative 'a'
so cosine of 'a'... if we say that this side is the adjacent side
this is the hypotenuse
this would still be the hypotenuse, right
and this is the opposite
and this is the negative opposite
so cosine of minus 'a' is equal to what?
this is minus a.. this is the adjacent over the hypotenuse
so it equals the adjacent over the hypotenuse... which we just say is 'h'
that's the same thing as cosine of 'a' right?
cosine of 'a' is also the adjacent over the hypotenuse
adjacent over hypotenuse
oh... I'm almost out of time
let me switch to a new video