Uploaded by khanacademy on 28.10.2007

Transcript:

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