i’ve already made a handful

of videos that covers what I’m going to cover, the

trigonometric identities I’m going to cover in this video. The reason why I’m doing it is

that I’m in need of review myself because I was doing some

calculus problems that required me to know this, and I have

better recording software now so I thought two birds with one

stone, let me rerecord a video and kind of refresh

things in my own mind. So the trig identities that I’m

going to assume that we know because I’ve already made

videos on them and they’re a little bit involved to remember

or to prove, are that the sine of a plus b is equal to the

sine of a times the cosine of b plus the sine of b

times the cosine of a. That’s the first one, I assume,

going into this video we know. And then if we wanted to know

the sine of– well, I’ll just write it a little differently. What if I wanted to figure out

the sine of a plus– I’ll write it this way– minus c? Which is the same thing

as a minus c, right? Well, we could just use this

formula up here to say well, that’s equal to the sine of a

times the cosine of minus c plus the sine of minus c

times the cosine of a. And we know, and I guess this

is another assumption that we’re going to have to have

going into this video, that the cosine of minus c is equal

to just the cosine of c. That the cosine is

an even function. And you could look at that by

looking at the graph of the cosine function, or even at

the unit circle itself. And that the sine is

an odd function. That the sine of minus

c is actually equal to minus sine of c. So we can use both of that

information to rewrite the second line up here to say that

the sine of a minus c is equal to the sine of a times

the cosine of c. Because cosine of minus

c is the same thing as the cosine of c. Times the cosine of c. And then, minus the sine of c. Instead of writing this,

I could write this. Minus the sine of c

times the cosine of a. So that we kind of pseudo

proved this by knowing this and this ahead of time. Fair enough. And I’m going to use all of

these to kind of prove a bunch of more trig identities

that I’m going to need. So the other trig identity is

that the cosine of a plus b is equal to the cosine of a– you

don’t mix up the cosines and the sines in this situation. Cosine of a times

the sine of b. And this is minus–

well, sorry. I just said you don’t mix it

up and then I mixed them up. Times the cosine of b minus

sine of a times the sine of b. Now, if you wanted to know what

the cosine of a minus b is, well, you use these

same properties. Cosine of minus b, that’s still

going to be cosine on b. So that’s going to be the

cosine of a times the cosine– cosine of minus b is the

same thing as cosine of b. But here you’re going to have

sine of minus b, which is the same thing as the

minus sine of b. And that minus will cancel that

out, so it’ll be plus sine of a times the sine of b. So it’s a little tricky. When you have a plus sign

here you get a minus there. When you don’t minus

sign there, you get a plus sign there. But fair enough. I don’t want to dwell on that

too much because we have many more identities to show. So what if I wanted an

identity for let’s say, the cosine of 2a? So the cosine of 2a. Well that’s just the same thing

as the cosine of a plus a. And then we could use this

formula right up here. If my second a is just my b,

then this is just equal to cosine of a times the cosine

of a minus the sine of a times the sine of a. My b is also an a in this

situation, which I could rewrite as, this is equal to

the cosine squared of a. I just wrote cosine of a times

itself twice or times itself. Minus sine squared of a. This is one I guess

identity already. Cosine of 2a is equal to the

cosine squared of a minus the sine squared of a. Let me box off my identities

that we’re showing in this video. So I just showed you that one. What if I’m not satisfied? What if I just want it

in terms of cosines? Well, we could break out

the unit circle definition of our trig functions. This is kind of the most

fundamental identity. The sine squared of a

plus the cosine squared of a is equal to 1. Or you could write that–

let me think of the best way to do this. You could write that the sine

squared of a is equal to 1 minus the cosine

sign squared of a. And then we could take this

and substitute it right here. So we could rewrite this

identity as being equal to the cosine squared of a minus

the sine squared of a. But the sine squared of

a is this right there. So minus– I’ll do it

in a different color. Minus 1 minus cosine

squared of a. That’s what I just substituted

for the sine squared of a. And so this is equal to the

cosine squared of a minus 1 plus the cosine squared of a. Which is equal to–

we’re just adding. I’ll just continue

on the right. We have 1 cosine squared of a

plus another cosine squared of a, so it’s 2 cosine

squared of a minus 1. And all of that is

equal to cosine of 2a. Now what if I wanted to get

an identity that gave me what cosine squared of

a is in terms of this? Well we could just

solve for that. If we add 1 to both sides of

this equation, actually, let me write this. This is one of our

other identities. But if we add 1 to both sides

of that equation we get 2 times the cosine squared of a is

equal to cosine of 2a plus 1. And if we divide both sides of

this by 2 we get the cosine squared of a is equal to 1/2–

now we could rearrange these just to do it– times 1

plus the cosine of 2a. And we’re done. And we have another identity. Cosine squared of a, sometimes

it’s called the power reduction identity right there. Now what if we wanted

something in terms of the sine squared of a? Well then maybe we could go

back up here and we know from this identity that the sine

squared of a is equal to 1 minus cosine squared of a. Or we could have

gone the other way. We could have subtracted sine

squared of a from both sides and we could have gotten–

let me go down there. If I subtracted sine squared of

a from both sides you could get cosine squared of a is equal

to 1 minus sine squared of a. And then we could go back into

this formula right up here and we could write down– and I’ll

do it in this blue color. We could write down that the

cosine of 2a is equal to– instead of writing a cosine

squared of a, I’ll write this- is equal to 1 minus sine

squared of a minus sine squared of a. So my cosine of 2a is equal to? Let’s see. I have a minus sine squared

of a minus another sine squared of a. So I have 1 minus 2

sine squared of a. So here’s another identity. Another way to write

my cosine of 2a. We’re discovering a lot of ways

to write our cosine of 2a. Now if we wanted to solve for

sine squared of 2a we could add it to both sides

of the equation. So let me do that and I’ll

just write it here for the sake of saving space. Let me scroll down

a little bit. So I’m going to go here. If I just add 2 sine squared

of a to both sides of this, I get 2 sine squared of a plus

cosine of 2a is equal to 1. Subtract cosine of

2a from both sides. You get 2 sine squared of a is

equal to 1 minus cosine of 2a. Then you divide both sides of

this by 2 and you get sine squared of a is equal to 1/2

times 1 minus cosine of 2a. And we have our other discovery

I guess we could call it. Our finding. And it’s interesting. It’s always interesting

to look at the symmetry. Cosine squared– they’re

identical except for you have a plus 2a here for the cosine

squared and you have a minus cosine of 2a here for

the sine squared. So we’ve already found a

lot of interesting things. Let’s see if we can do

anything with the sine of 2a. Let me pick a new color

here that I haven’t used. Well, I’ve pretty much

used all my colors. So if I want to figure out the

sine of 2a, this is equal to the sine of a plus a. Which is equal to the sine of a

times the co– well, I don’t want to make it that thick. Times the cosine of a plus–

and this cosine of a, that’s the second a. Actually, you could

view it that way. Plus the sine– I’m just

using the sine of a plus b. Plus the sine of the

second a times the cosine of the first a. I just wrote the same thing

twice, so this is just people to 2 sine of a, cosine of a. That was a little bit easier. So sine of 2a is equal to that. So that’s another result. I know I’m a little bit tired

by playing with all of these sine and cosines. And I was able to get all the

results that I needed for my calculus problem, so hopefully

this was a good review for you because it was a

good review for me. You can write these

things down. You can memorize them if you

want, but the really important take away is to realize that

you really can derive all of these formulas really from

these initial formulas that we just had. And even these, I also have

proofs to show you how to get these from just the basic

definitions of your trig functions.

Any chance of a video on the other trig identities for sec(x) etc ?? Loved this one videos always help me learn 🙂

which software are you using?

"Two birds with one stone"….

Who invented that expression? :/

Sounds cruel…

Primitive bird hunter

Yeaaaah, I shoulda watched the earlier videos, I have no idea what hes talking about

@Thymonico he's actually killing too hunters with one IQ test. hunters generally aren't very bright.

@the1nonly12507 That would take forever…

To all who are posting, I believe that the main reason for why Sal is not really responding to any of your questions and comments is because he has too many videos online to respond back and review…just a thought.

@jus400track I try though, but, yes, it is getting difficult now

See? Teaching stuff really isn't that hard.

Most teachers just act like you're an idiot if you can't understand them.

@idster7 You mean two right? Also, have you ever met a hunter who wasn't bright? I honestly haven't, and being a hunter I've met plenty. But I have met plenty of dim prejudists (I know that's not a word but you get the idea)

@JSnyder49428 yes, i meant two*. intelligence & empathy toward animals are correlated. i'm not saying hunters tend to be stupid. they're independent, perhaps contrarian, people. they may be smarter than the average. but very smart people tend not to be hunters. what percentage of college professors or macarthur grant winners do you think are hunters? a far less percentage than the general public, and not just cause they're too busy.

@idster7

1) I would like to see your source for intelligence & empathy being correlated, it's not a given, as a matter of fact, I'm a psych major right now and I'm skeptical.

2) Empathy toward animals and hunting (against simple logic) aren't mutually exclusive. Again, I take myself for example, I hunt, and I volunteer for the ASPCA from time to time. Hunting is more about being in nature and preventing overpopulation than it is killing and shooting stuff.

ran out of room

@JSnyder49428

3) You aren't from MI are you? 3 of the 5 professors I have right now are hunters. BUT, don't you think there are a plethora of other factors influential on whether or not people hunt? Take location for example, how many universities are near hunting grounds? I could go on and on about reasons like this…

4) How do you determine "intelligent people"? by the awards they've received? the subjects they teach? How successful they are? And I ran out of room again

Respond to this video…

I think the best method so far is IQ testing, and even that's debatable at best. Actually, college professors aren't unusually intelligent people, most of them are just average people who decided they like teaching.

salll iss amaazingg!

Aren't you supposed to use double angle formulas?

@khanacademy It's ok! More videos are more helpful than answering comments 🙂

thanks. using your videos to review for my test while I carb-load before the test!

i'm so in love with you right now!!!

It's too bad that Khan can't do a video on how to write papers. I'm struggling with mine…

cos2a=(1-sin^2a)-sin^2a

wouldn't you expand the brackets? why did you just drop the brackets and subtract?

@dragonpaint37

The brackets are being raised only to the power of one. Anything raised to the one power is itself.

(1 – sin^2 a) – sin^2 a

1 – sin^2 a – sin^2 a

1 – 2sin^2 a

@collcool0 actually anything raised to the power of zero is one. Anything raised to the power of 1 is itself. However, in this case, 1-sin^2 a is 1 because it is a trig identity.

@dragonpaint37 because it's a minus sign that follows the bracket, not a multiplication sign… therefore you do not distribute…

watch the whole playlist, honestly, all of these videos are made in a progressive way, so you would need to watch all the videos before this in the same playlist.

sal..just love u…

god bless you..

lol

2 people couldn't find the fun

I demand my 25 years back to restart again with what I know now!

did you know that my school doesn't have a english teacher? thanks MR KHAN

Thank you

"fun" god dammit sal

4 people failed Trig.

can anyone help me understand the third line?

this isn't fun

My brain is ded.

Thank you for showing me how all the identities connect together. This was extremely helpful.

This shit is so hard… This alone will probably make me fail my math test >.>

"Fun"

If this video didn't exist, I may have committed seppuku trying to learn this shit

amazing! all looks so easy and nice! But by the time O try to put into my homework is just awful.I think I will cry.

I think you lost me at 0:31… O_O

how do we know the first one?

I mean… The math is pretty straight forward, but how on earth do quickly gain an overview of this? That Khan can effortlessly derive all these formulas, keeping track of which have already been derived, and how he can use them to find new ones, is amazing to me!

"fun"

I don't understand the difficulty? This video is extremely and straight forward and helpful. These trigonometric identities are super helpful in solving Physics problems. Learn and remember them now!

take a shot every time he says sin

i don't understand

WOW, your voice in is the Khan PHYSICS videos too!!!

Is he a genius professor or sth?

God bless you

i'm going to crazy

What did he mean by this

went back and watching the other law of sines video, you never once mention that first identity which you say we should now. utter crap

rip

you lost me somewhere ;–; come back (~ ;_>;)~

luckily its already in the formula sheet for alevels

Math is a beautiful subject. To better get everything about trigonometry and others, check out this app https://play.google.com/store/apps/details?id=com.gamecodeschool.mathtoolkit2

Awesome and clear explanation .

Cos and sec can be converted into positive if their theta value is in negative. And for other ratios it is not applicable.

you are bad at teaching

This was absolutely exceptional. I love that feeling when your teachers tell you to memorise a formula without really telling you why, and then after you find out the proof you have the "ohhhh" moment.

Thank you sooo much!!!

Very good review amen brother

Does anyone know what drawing program is used in these videos?

it's summury video. Watch the previous videos to understand this.

Thanx sal for the revision.

does smoking weed and calculus go together…..?

How cos 2cos square a-1=cos 2a

THANKS

KHAN U HELPED ME A LOT

Excellent review.

two birds.. My. brain is one of them birds.

you lost me at 2 birds 1 stone.

oh waitThank you so much sal and all those who work effortlessly to make these vids happen!! Love you all. Its a sad time in education when educators like my teachers cant satisfy my questions on where these came from, but on the other hand there are educators such as yourselves that actually care about spreading understanding instead of just making little robots that can spit out some answers.

LOL I almost thought the video ended at 6:55!!

lmao sitting here at 11 pm on a Friday night preparing for my NBT tomorrow..

I have a test on this today and we only had 2 classes to learn the whole unit, wish me luck 😛

Pretty sure I didn’t have fun

pretty sure the first one is messed up. Its actually sinacosb+cosasinb

Where’d he get cosine from in the beginning?

Correct me if I’m wrong, bur shouldn’t there be parenthesis around the “sin(a)cos(a)” on the last problem at 11:06

Hopefully i can finally pass my calculus class with this knowledge

RIGHT BEFORE MY FINAL EXAMS

0:47 so you're going to assume I actually know something?

Harsh

I love the timbre of your voice, sire

i hope i can pass my precal midterm tomorrow