help-36

[mossy inlet]

well my class did this like last week

[mortal island]

HOW

[tepid valley]

You know the formula for cos2x?

[mortal island]

LITERALLY HOW

[mossy inlet]

now we're on like polar but we just did double angle formulas and stuff

sum and angle bla bla

[mortal island]

IM IN FUCKING CALCULUS

HOW ARE YOU AHEAD OF ME

[tepid valley]

??

[mortal island]

WHAT THE ACTUAL FUCK

[mossy inlet]

um

[mortal island]

🥲

[timber basin]

guys?

[mossy inlet]

i know calc too! but im not in calc at school

anywayyy off topic, reading problem a sec

[mortal island]

Sorry sorry

Side tracked

[timber basin]

lols nw.

[tepid valley]

Do you know the formula for cos2x????

[mortal island]

Gimme like…2 minutes I’ll brb

[tepid valley]

:/

[timber basin]

okay, we have an expression with a numerator that looks like (1-cos(2theta))

[timber basin]

hey, wanna take it from here?

[tepid valley]

I don't mind.

As long as you're okay.

[timber basin]

thanks 🙂

[tepid valley]

Okay then.

Well cos2x = cos²x - sin²x

[mossy inlet]

am i overthinking this...?

i solved it in like... 3 lines...

[timber basin]

true, but also cos(2x)=1-2sin^2x by extension

[tepid valley]

It's a standard formula, try to keep this in mind.

[tepid valley]

Yeah...

[timber basin]

ur not

[tepid valley]

Yup!

[mossy inlet]

^^ i used this one for the top, and 2cos^2x-1 on the bottom

[mortal island]

Hold on chat

I’m not even looking at this

[tepid valley]

Yes.

[timber basin]

ditto

[mortal island]

I’m trying to check my BP

I have blood pressure issues that flare when I get overwhelmed

[tepid valley]

Pretty cool thing but wherever you see 1-x or 1+x just put cos2theta in place of x.

[tepid valley]

Cuz this is so darn useful.

[tepid valley]

Especially while doing integration.

[mossy inlet]

oh i see...

i haven't gotten far enough in my integration studies to know that tidbit, thats cool

[mortal island]

I hate that I’m so bad at math that I have the active role now

[tepid valley]

Yeah now you know.

[mossy inlet]

lmaoo i have active too

[timber basin]

Here's nr. 2 (recap)

[mortal island]

Yeah well you’re good at math

[mossy inlet]

nahh i struggle too dwdw ive opened plenty of help channels

[mortal island]

I’m only active here because I don’t know anything

[tepid valley]

Why do I not have it yet lol, I've sent like 300 messages. 😭😭

[mossy inlet]

lemme check how many i sent

[timber basin]

first step to mastery..?

[mortal island]

I’ve been in this server like all day everyday for the past 2 weeks

[mossy inlet]

oh... uh.

[mortal island]

Probably not

[mossy inlet]

welp. perhaps more than 300.

[tepid valley]

💀

When did you get it?

[mossy inlet]

i have a life i swear 😭

hmm lemme think

[tepid valley]

It's a good thing, your dedication is really cool.

[tepid valley]

Eh it's a math server.

Not a bad thing.

You're spending your time here.

[mossy inlet]

humm

probably around 3/22/26 igs

[mortal island]

In this channel alone I have over 700 messages

🫩

[tepid valley]

I'm crine

[mortal island]

It’s not dedication I’m just so behind from being sick that I can’t catch up

[mortal island]

Not helpful

[mossy inlet]

lmaoo its okay growth isn't linear and imporvement is improvement

[tepid valley]

Why can't you catch up?

What's holding you back?

[mortal island]

She keeps pulling me out of class to do tests I didn’t learn the material for which takes me out of class for more material

[tepid valley]

Hmm.

[mortal island]

So she pushes back my tests to give me a chance to learn the material that i can’t learn because she doesn’t teach me

[tepid valley]

What grade are you in?

[mortal island]

And the cycle repeats over and over and over again

[mossy inlet]

did she provide you with like

[tepid valley]

I can try recommending books.

[mossy inlet]

maybe

[mortal island]

I’m a junior

[tepid valley]

To help you cover stuff.

[mossy inlet]

practice problems or like notes or something?

[mortal island]

She does but I don’t understand any of it

And none of my classmates like me enough to help me

[tepid valley]

Junior means what age sorry.

Age****

[mortal island]

And I don’t have time to go talk to her because I have class during her one free period

[mortal island]

Im 17

[tepid valley]

Uh what's your portion like.

[mossy inlet]

does your school offer free tutoring/after or before school help?

[mortal island]

No

This assignment is like 2 months overdue because she keeps assigning shit idk how to do and I can’t do it

[mossy inlet]

oh

umm

[tepid valley]

It's okay bro don't fret too much.

Just be patient.

And learn math slowly and stepp by step.

Slowly*

[mossy inlet]

you can try khan academy or watching yt videos like organic chem tutor or others

[mortal island]

I have 2 days to get 2 proof assignments done and one asymptote assignment or I fail the class

[tepid valley]

Yikes.

[mossy inlet]

Sooo with proofs you have to keep the end goal in mind, asymptotes, ummm what asymptotes? like with trig or?

[mortal island]

Hold on I can’t do this rn

I can’t breathe or see

[tepid valley]

Alr just try to do the assignments for now.

After that learn maths properly.

[mortal island]

That’s what I’ve been trying to do

[tepid valley]

See idk about your portion.

But like I can recommend sm called NCERT.

It's a very simple book for math basics.

[mortal island]

Idk what you’re talking about

[timber basin]

Hey Milo, I have to leave now, but i just wanted to say that I think it's great that you're taking the initiative to learn this kinda stuff even though you don't like math. It sucks when you have to learn and feel like you're behind everyone, and uhm, respectfully speaking, the other people talking about their own progress is not helping that. It takes a while to do math, and it feels like no progress is done sometimes, and it's okay. Just hang in there alright? Take some breaks when you need them too. Believe in ya! 🙂

[tepid valley]

For 12th grade.

[mossy inlet]

idk what it is either but if its a book u can probably find it online

[tepid valley]

Yeah the book is called NCERT.

It's in the Indian curriculum.

[mortal island]

That’s not gonna help me

[timber basin]

I'd just focus on the assignments first, then potentially if you want you can go to the reading.

[tepid valley]

Yeah exactly.

First assignments.

[mortal island]

I can’t focus on anything rn

[tepid valley]

Then do the books.

[mortal island]

I can’t see or breathe or anything

[tepid valley]

2 days is plenty of time.

Just relax.

You got this.

[mortal island]

I’m trying

It’s easier said than done

[tepid valley]

Use chatgpt or whatever to get this done for now.

[timber basin]

Yeah! You got it. Take a step outside and a breath of fresh air. Then you can tackle it again.

[tepid valley]

I'd not recommend it but like this is the last resort.

[mortal island]

I can’t do that I have to turn this in for her to grade

Chat gpt doesn’t show the steps

She’s grading the steps not the final answer

[tepid valley]

I'd have helped you do the assignment out but I got a big exam on the 15th myself.

[mortal island]

I’m trying to collect myself and figure this out

[mossy inlet]

alrighttt so i would def recommend just taking a break and then like, i can try to help and stuff

[mortal island]

Ok thank you

[tepid valley]

Why don't you send the assignment for now?

[mortal island]

Ok

One sec I’m not at my computer

[tepid valley]

Np.

I'll go brush my teeth or sm, I just woke up. 🥀

[mortal island]

It doesn’t help that it’s late

I’m also gonna go change clothes real quick

[tepid valley]

Btw believe me or not I'm in a similar situation.

I barely know anything in organic chemistry.

And I have a huge ahh exam which has oc in it.

I'm trying out what I told you to do- just patiently learning it.

[mossy inlet]

damn

i had an fbla competition and missed one day of math and i have math tues and i have to have my pages of homework done, but i dont understand what a limacon is or why those flowers form. so i guess im also a lil cooked lwk (i hate polar-)

[tepid valley]

😭

What's fbla

[mossy inlet]

future business leaders of america (club)

[tepid valley]

Oh damn.

[mossy inlet]

had states so i missed fridays math lesson

[mortal island]

I think I have an intellectual disability

[mossy inlet]

like i really dont care what a dimpled limacon is-

[tepid valley]

My exam is like life deciding.

I'll get my college based on this.

[mossy inlet]

i swear in all my studies of calc ive never had to use a limacon so does ts matter

[tepid valley]

🥀

[mossy inlet]

nuuu its okok you got this

[tepid valley]

Oh.

[mortal island]

Time to lock in on organic chemistry tutor

[tepid valley]

Nah... My exam requires much more depth than what he teaches...

[mossy inlet]

(if you're serious) have you tried reaching out to counselors? (if you're not serious) don't beat yourself up, its okay

[tepid valley]

Just memorizing won't help.

[mortal island]

I’m half serious half not-

[modest sequoia]

hi milo

[mossy inlet]

its alright... i get it, dont worry you'll get throuhg it

[mortal island]

KRISH MY SAVING GRACE

[modest sequoia]

😭 😭

[tepid valley]

Yea

[modest sequoia]

im doing my own math hw rn lmao

[mortal island]

YOU COULDNT HAVE SHOWN UP BEFORE I HAD A PANIC ATTACK

[mossy inlet]

lmao i need to do my roses and limacons assignment- fahhh

perhaps i shall make my own help channel and beg someone to teach me ts

[modest sequoia]

i needed to take a nap i was way too tired and then realized my hw is due tomorrow

[mortal island]

Oh shit

[tepid valley]

Yoo send dat assignment?

[mortal island]

I’m sure in reality it’s not that hard

[tepid valley]

Oh wait this was the assignment?

[mortal island]

And it’s just late and it’s been a while since I’ve done this

[modest sequoia]

oh i love proofs lol

[mortal island]

Yeah

[tepid valley]

Is this the entire thing?

[mortal island]

No

There’s more

[tepid valley]

Send the pdf file then.

[mossy inlet]

so number 3 now right?

[tepid valley]

Of the entire thing.

[modest sequoia]

^do not send a pdf file. send screenshots of it

[mortal island]

No-

[tepid valley]

Bro I'm not gonna lie.

[mortal island]

I wasn’t gonna send a pdf

[tepid valley]

You just gotta memorise like 10-20 formulas for this one.

And you'll be able to do these.

[mossy inlet]

oh

[mortal island]

I have the formulas beside me

[mossy inlet]

nvm

i thought we were on number 2 earlier

[mortal island]

We were

[tepid valley]

And they are straightforward applications of the formulad.

[mortal island]

We never got done with number 2-

[mossy inlet]

so looking at like the end result i would think like which formula would be best

[modest sequoia]

just making sure cause May said so lol. for future reference @tepid valley we don't like people sending files of any sort in case of viruses or anything of the sort that could possibly be on there

[mortal island]

That’s what I was trying to tell him earlier and then I got confused cus he was doing things weird

[tepid valley]

Oh didn't know, thanks.

Then send photos of all the pages. :p

[tepid valley]

Yeah.

[modest sequoia]

nw, a lot of times screenshots work well enough or if it's a picture and its unclear we just ask to take a better pic lol

[mortal island]

Ok let me find em on my phone so it’s easier

[mossy inlet]

heres my work for number 2

so basically like

[mossy inlet]

i saw that in the end result on the RHS

[modest sequoia]

.pin

[mossy inlet]

there was a sin squared in the top and a cos squared in the bottom

[mortal island]

Ok- can you walk me through that from the beginner

[mossy inlet]

so i plugged in the cos2x formula that would work best (having those terms)

[mortal island]

Beginning*

[mossy inlet]

so cos(2x)=1-2sin^2x right?

(as one of the many forms it can take on, of course)

[mortal island]

Hold on I’m not following

[tepid valley]

It's a formula.

You gotta memorise it.

[mortal island]

No I’m just lost at where we are

[mortal island]

And what part of the equation we were on

In the first place-

[tepid valley]

The numerator of lhs

[mortal island]

I have the formulas guys 🥲 I’m just not- all there rn

[pine whale]

https://www.youtube.com/watch?v=0ggGQ96eaiE

[mossy inlet]

okay so like let's think just about the top half of the left side. and plug in for cos(2x) in a way that would keep a cos^2x in the top like the RHS

[mortal island]

Guys the formulas is not my issue I promise you 😭😭😭😭

[tepid valley]

Ye we got that dw.

See is LHS there is 2x everywhere.

In RHS there is x not 2x

[mortal island]

So…. 1-2sin^2x?

[tepid valley]

So we gotta convert the 2x into x somehow.

[tepid valley]

That's why we did this.

[tepid valley]

Yea.

[mossy inlet]

yup

(i said cos^2 earlier i meant sin^2 mbmb)

now, apply the same thought to the bottom of the LHS

[mortal island]

I figured that’s what you meant

I would do…. Cos^2x - sin^2x I think-

[mossy inlet]

hmm.. good thought. is there one thats better, perhaps with no sin in it and just cosine?

[mortal island]

The 2cos^2x one?

But how do you get rid of the 2-

[mossy inlet]

mmhm cus thatd make it easier to simplify later right?

[mossy inlet]

one step at a time..

plugging back in... remember the original ones

plug in the formulas for cos2x

and see how you can simplify that

[tepid valley]

Alr then ima go study some OC, ping me if y'all need help!

All the best evelyn and milo.

[mortal island]

Thank ya

[mossy inlet]

glgl may

[mortal island]

Which ones-

[mossy inlet]

er

the ones you found

like

the one you picked

[mortal island]

There are multiple that could theoretically work in this case

[mossy inlet]

for the top and the bottom

[mortal island]

That apply to cos2x

[mossy inlet]

^^ for the bottom

[mossy inlet]

^^ for the top

[mortal island]

Right

[warm radish]

Help me bro

🙏

[mortal island]

Go find another help channel

[warm radish]

Hgf6uhg

[mortal island]

This one is currently being used

[mossy inlet]

this is occupied... also this isn't even a problem this is just random pics lol

[mortal island]

And there are about 5 other open channels rn for yourself

[mortal island]

I was gonna say

None of that makes sense lmao

[mossy inlet]

lmao its probably a generic math pic

[final saddle]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[mossy inlet]

lol

[mortal island]

I think this guys a troll anyways

LMAO

[mossy inlet]

anyway off topic

[mortal island]

Anyways…

[mossy inlet]

back to plugging it in...

what do you get? with the formulas you picked

[mortal island]

I’ve got 1- (1-2sin^2x)/1+(2cos^2x)

[mossy inlet]

remember the -1 for the formula in the bottom

should be 1+(2cos^2x-1) but close

[mortal island]

Huh…

Oh I see

Fixed

[mossy inlet]

So now...

how would you simplify this 🤔

what could cancel...

[mortal island]

It looks like a lot of it should just…cancel

The 1’s on the outside

[mossy inlet]

yesyes on the right track

[mortal island]

And the 2’s?

And the 1’s on the inside?

[mossy inlet]

yup...

[mortal island]

?

[mossy inlet]

like

uh

lemme write it out

[mortal island]

So there are steps involved 😔

[mossy inlet]

so heres the first step in going from the expression you built to the RHS

so i distributed the negative sign in the top

and reomved parentheses in the bottom

so far so good?

[mortal island]

If you distribute the negative shouldn’t that first 1 go away?

[mossy inlet]

Yup, it'll cancel

(left it like that so you could notice, and also for baby step work)

now on the bottom...

less noticeable perhaps, but, anything similar popping up?

[mortal island]

There’s a 1-1 on top and a 1 + (-1) on bottom?

[mossy inlet]

mmhm

and those both cancel to...

[mortal island]

0

[mossy inlet]

yup!

so now the new expression is...

[mortal island]

2sin^2x/2cos^2x

[tepid valley]

Hi guys.

[mossy inlet]

yes!

[tepid valley]

Oo still on first one.

[mossy inlet]

and now how could we further simplify this... hmmmmm

[tepid valley]

2nd*

[mossy inlet]

yup been working thru the steps

[mortal island]

2/2 cancels to 1?

[mossy inlet]

so now we are left with...

[tepid valley]

I see I see.

[mortal island]

Sin^2x/cos^2x…..? With a 1 somewhere….

[mossy inlet]

err

well the 1 is like

invisble

sorta

[tepid valley]

1 × anything = anything

[mossy inlet]

^^ yup this, the identity property

[warm radish]

I am the lorax. I speak for the trees. For the trees have no tongues.

[mortal island]

Ok- so just sin^2x/cos^2x

[tepid valley]

Hi the lorax

[mossy inlet]

.... ||srsly gng, go to #chill ||

[mortal island]

Ok this dudes just a troll

[modest sequoia]

<@&268886789983436800> troll

[mossy inlet]

Righttt!

And that's what we were looking for, right?

[tepid valley]

Noice.

[mossy inlet]

We wanted the left hand side to be the right

and now it is

after we algebraically manipulated it, its equal

[tepid valley]

At the end of everything write LHS = RHS
Hence proved.

Btw Milo did you get the answer to 1?

[mortal island]

Yes

[stray hill]

Do not troll in help channels, keep shitposting in #chill

[warm radish]

Trees are no joke man

[mortal island]

Ok so number 3…I’m thinking

[tepid valley]

Evelyn come back. 😭

I'll help since I'm here anyways ig.

So in the first one.

[mortal island]

If you can help idm to work w you

Hold on chat lemme think

I wanna give an attempt

[tepid valley]

You remember how you got sin2x in the first one?

[tepid valley]

Bet.

[modest sequoia]

lmk if you dont get how to explain something im active just doing my hw alongside lol

[mossy inlet]

lol sorry i was helping in another channel

[mortal island]

Krish explains it in dumb dumb terms for me

[mossy inlet]

while doing my math, chinese, and bio homework

[mortal island]

I can do bio

[tepid valley]

Thanks lol I think I suck at explaining but I'll try my best.

[mortal island]

I’m good at that

[tepid valley]

Multitasking final level. 🥀

[mossy inlet]

and doomscrolling

um...

[mortal island]

OK IDEA

I think….i think i wanna take 2sin^2 x and turn it into…

Cos2x-1

Wait…that doesn’t seem right

[tepid valley]

Okay then?

[tepid valley]

Just trust the process in math. 🥀

You'll always get the right answer somehow.

[mortal island]

No I mean I don’t think i manipulated the formula correctly

[tepid valley]

Yeah you didn't but you're close.

[mortal island]

Cus it’s cos2x= 1-2sin^2x

[tepid valley]

It's 1-cos2x

[mortal island]

THATS IT

[tepid valley]

Alright then?

[mortal island]

Lemme write it down

Right ok…

Uhmmmmmm

Then I wanna turn…cot^2x to csc^2x -1

Which gives me 1-cos2x +csc^2x -1

So I can cancel the 1’s

And then move the negative cos2x to the other side of the csc

And it’ll be equal

@tepid valley does that seem right?

[tepid valley]

Good thing you pinged I closed dc. 💀

Yeah anyways.

[mortal island]

YESSS

[tepid valley]

Wait I'm reading.

[mortal island]

Oh

Here I’ll take a pic of the work too

[tepid valley]

Yea thanks.

[pine whale]

,rotate

[tepid valley]

Whoa- where did you get 2sin²x + cot²x?

[mortal island]

The….original problem?

[tepid valley]

The question is 2sin²xcosx right?

[mortal island]

Not number 3

[tepid valley]

Oh.

[mossy inlet]

uh

[mortal island]

Wait nvm

I did number 4

[mossy inlet]

i think number 4

yeah

[mortal island]

And labeled it 3

[mossy inlet]

oh well

[tepid valley]

Yeah that's right.

[mortal island]

Well is it at least right

YESSS

Just gonna…relabel it

[tepid valley]

This is perfect.

Here's some tips tho- write every equality on a diff line pls. 😭

[mortal island]

Fair enough I’m just trying to save paper

I only have this one page atm 😭😭😭😭

[tepid valley]

Is this rough?

[tepid valley]

Get more, all this is not gonna fit in a page bro.

[mortal island]

Oh I know

But it’s all in my locker rn

I’m just trying to get through 5 tonight

And then I’m gonna go to bed

Cus it’s 11 pm

[tepid valley]

https://tenor.com/view/son-im-crine-son-im-crine-gif-5176045023480767429

[mortal island]

Ok actually for number 3 this time

[tepid valley]

Alright then.

[mortal island]

I wanna do the same thing

[tepid valley]

Mhm.

Go ahead.

[mortal island]

1-cos2x

Does that combine at all with the other cosine there?

[tepid valley]

Btw this is how catchpoints work.

[mortal island]

What’s a catch point

[tepid valley]

What you're doing is wherever you see 2sin²x, you're using this formula.

[tepid valley]

It's not an official word but my teacher uses this.

[mortal island]

Ag

[tepid valley]

So yeah good.

[mortal island]

My teacher doesn’t believe in subtraction so-

I get weird things

[tepid valley]

??

[mortal island]

Or division

[tepid valley]

Wha

Anyways just do the problem.

[mortal island]

Yeahhhh…she’s weird

[mortal island]

^^

[tepid valley]

Yea you'll get (1-cos2x) × cosx

What after this tho.

[mortal island]

Ohhh ok

Hmmm…how about changing sin2x into 2sinxcosx

[tepid valley]

Nice.

NICE*

But the thing is.

[mortal island]

?

[tepid valley]

Where is sin2x?

You have 2sinxcosx

You gotta turn that into sin2x

In step 1.

[mortal island]

No on the RHS

[tepid valley]

Oh I see.

Sure do that.

What do you getM

[mortal island]

2sinxcosx/cscx

[tepid valley]

Perfect.

So this time instead of making the LHS the RHS

You made the RHS the LHS.

Which is absolutely okay

[mortal island]

They don’t match tho…

[tepid valley]

What's cosec x?

[mortal island]

Uhhhh 1/cosx?

[tepid valley]

Nope 1/sinx

[mortal island]

Oh yeah

[tepid valley]

sec x is 1/cos

[mortal island]

Yeah

[tepid valley]

Bro see.

In trigo.

We do discrimination.

We don't give a f about sec cosec and cot.

Always turn them into cos sin and tan.

[mortal island]

Ok but they still don’t match 😭😭

[tepid valley]

What did you get?

[mortal island]

Here’s where I’m at

[pine whale]

,rotate

[tepid valley]

Well see.

First of all whenever we're doing proofs.

You can't start by saying they are equal.

[mortal island]

I’m…not

[tepid valley]

Like you can't start with 2sin²xcosx = sin2x/cosecx

[mortal island]

That’s the problem….

[tepid valley]

This means they are equal.

See start like this.

[mortal island]

Go look at it

[tepid valley]

Oh wait.

[mortal island]

I literally wrote what was on the assignment

[tepid valley]

Bro but still.

[mortal island]

Wdym?????

[tepid valley]

How did you get the second line.

You got that from the first line right.

[mortal island]

Changed 2sin^2x into 1-cos2x

And changed sin2x into 2sinxcosx

[mortal island]

Yeah

[tepid valley]

Yea that's the thing.

The question isn't saying those 2 are equal.

We have to show they are equal.

[mortal island]

I know

[tepid valley]

You cannot write the first line.

[mortal island]

?????

[tepid valley]

And without the first line you can't get the second line.

[mortal island]

Dude it’s just there to show which side is which

That’s how I’ve been doing all of these

[tepid valley]

Ik but that's not how it works.

[mortal island]

But it is????

[tepid valley]

You gotta start with something like.

[mortal island]

I can’t change the original problem

That’s what the original problem is

[tepid valley]

I'll show you the 2nd question.

The way to write the steps.

[mortal island]

It’s the same on the second question

[tepid valley]

Give me a second.

[mortal island]

And the 4th

And the 1st

Everything is set equal to something else

I know that we’re proving they’re equal

I know they’re not rn

But there has to be a sign there

[tepid valley]

See what I did here?

[mortal island]

No

[tepid valley]

See the steps are all correct in all your qn.

I'm just showing the way of writing.

[mortal island]

I’m just doing it the way my teacher taught me

[tepid valley]

Fine yk what I think this'll confuse you more.

Let's just carry on with the questions.

We'll talk Abt this later.

[mortal island]

I can show you my notes on it if you want 😭😭😭

Directly from her

[tepid valley]

Sure.

But later.

First let's get these questions done.

[tepid valley]

Frame them all like your teacher wants you to later.

[mortal island]

I’ll get a better pic of the top part too

[mossy inlet]

um.. just came back from helping someone with a log question

[mossy inlet]

idk my teacher says we wont get the proof questions on the ap exam, so she just has us simplify to the RHS or whatever, like, just tells us simplify the LHS and try to get it equal to RHS

[mortal island]

Ignore how bad my hand writing is

[tepid valley]

First few lines are unclear.

[mortal island]

Wdym

[tepid valley]

Like blurry.

Send pic of that again.

[mortal island]

Oh

[tepid valley]

Yeah that's the thing.

This is da wae

[mossy inlet]

i mean the notes (we do guided premade notes not made by my teacher) that she had us cross out had like

an equals mark

with a question mark over it

and at the end they put a checkmark over it when they proved it

but idk if thats how its supposed to be done or not cus like well, we crossed them off cus those dont come on ap exam

[mortal island]

Yeah idk this is just how we do it

[tepid valley]

Hmm.

[mortal island]

We use a little gray box at the end to signify that the problem is complete

[tepid valley]

Basically if you have to prove sin²x + cos²x = 1

You can't say sin²x = 1 - cos²x

[tepid valley]

Yeah.

[mortal island]

Yeah

[tepid valley]

Oh I see what you're trying to do.

[mortal island]

It has to be sin^2x + cos^2x = sin^2x + cos^2x or 1 = 1

[tepid valley]

As long as you don't bring stuff from one side to other side.

Then you can do whatever you want..

[mortal island]

I guess..?

[tepid valley]

Alright anyways.

[mortal island]

Can I add you and we resume in the morning?

[tepid valley]

Make the cosec x = 1/sinx

[tepid valley]

Les finish the 3rd one.

It's very simple.

[mortal island]

Ok

[tepid valley]

Also idk if I'll be able to help.

I gotta do oc.

Ask evelyn.

She's a much better teacher than me. 🥀

[tepid valley]

Did you do this?

Oh btw if you didn't know.

a/b/c = ac/b

[mossy inlet]

🥀 im 14 im not that useful-

ill try

i finsihed my bio hw

[tepid valley]

I didn't even know double angle formula till I was like 15.

Wasn't in the syllabus till 16.

So you're already better. 🥀

[mossy inlet]

i learned it in this online thing my district had

when i was er

13?

before i turned 14 in 8th

[tepid valley]

Damn.

[mortal island]

You remind me of this kid in my calc class

[mossy inlet]

er... i had... yeah id say exponential progress xd i started but never finished unit circle in 7th. gave up on roses/limas in 8th (but finished most other trig work). learning calc on own this year

[mortal island]

He has no reason to be in there other than it’s a requirement

He’s doing shit not even the teacher knows

[tepid valley]

Damn, all the best ping me if you need help with something.

[mossy inlet]

lmao

[mossy inlet]

tytyy

[mortal island]

And he’s like….15? At best

[tepid valley]

Requirement..?

At 15?

Blud what.

[tepid valley]

Np, calc is very fun.

[mortal island]

No like he knows everything

He’s just in the class because it’s a requirement for graduation

[mossy inlet]

oh

yeah

fair

[mortal island]

He’s way past what’s happening

[mossy inlet]

according to the UC graduation req they gave us

[tepid valley]

Wish that was in my country too.

[mossy inlet]

i have 6 years of high school math

[mortal island]

He literally just sits in there and does his own level of math

[mossy inlet]

but according to my school

i have 1 semester

so i need to take a second year

to graduate

[mortal island]

Anyways goodnight guys I rly gotta get some sleep

[mossy inlet]

even though college admissions says i have like half a decades worthhhh ackkk i hate my schoo-

[tepid valley]

Goodnight punpun.

[mortal island]

Oh???

[tepid valley]

Uh nvm.

[mortal island]

What does that mean 😭😭😭😭

[tepid valley]

Nothing.

💀

[mortal island]

Sure…

[tepid valley]

Take it ez.

[mortal island]

Idk if you’re flirting with me or insulting me or…babying me

[tepid valley]

Not really.

Neither.

There's a comic called goodnight punpun so I remembered.

[final saddle]

If you are done with this channel, please mark your problem as solved by typing .close

[mortal island]

.close

[final saddle]

Channel closed

Closed by @mortal island

Use .reopen if this was a mistake.

[final saddle]

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• Type the command .close to free the channel when you're done.
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[static oak]

Could somebody explain this to me it’s from an answer key and I don’t understand it at all

[gritty chasm]

do you know the theorem that $\log_a(b) = \frac{\log_c (b)}{\log_c (a)}$

[soft zealot]

Annie Maqionde

[static oak]

No

[gritty chasm]

its called the Change of Base Formula.

[static oak]

OHHH

then yes I do

Ok this is easy then lol ty

For this one I’m not sure how to finish a and b

[gritty chasm]

do you know $\log(ab) = \log(a) + \log(b)$

[soft zealot]

Annie Maqionde

[gritty chasm]

and what is $\log(a^x)$?

[soft zealot]

Annie Maqionde

[static oak]

Yea

[static oak]

Xlog(a)?

I dknt know how to do those with a tho since it’s multiple in one

[gritty chasm]

use both of these.

[static oak]

yea I know I should but I don’t know how to combine them

Cus I want the ^3 to only impact the 2 not the 3

[gritty chasm]

first this

[gritty chasm]

then this.

[gritty chasm]

why is there a $\log 3 \log_5(2)$

[soft zealot]

Annie Maqionde

[gritty chasm]

its just $3 \log_5(2)$

[soft zealot]

Annie Maqionde

[gritty chasm]

remove the log in the front.

[static oak]

Whoops forgot I put that there

For 3 j don’t know what to do. I’m assuming I’m not doing change of basses since the base log has the base of 5 already. But I don’t know how to do multiplication cus no squared , ^3 etc will make 2 into ten. Or make 5•6

[gritty chasm]

i cant see 3j

[static oak]

Sorry I meant 3b

Idk why I typed j

[south dirge]

rewrite 30 into the suitable factors

as in, 30= 6x5=2x3x5

And then simple do what you did before

[static oak]

Yea but I have to make them into a and b and I can’t figure out how to make 2 and 3 into 6 and 5

like 2^4,2^3 is 8

[south dirge]

2 and 3 into 6?

They are js multiplied no?

[static oak]

3^2 is 9

[south dirge]

yeah so here you have 2^1 x 3^1

no need for exponents of 2 and 3

If you are confused as to what to do with 5,

observe closely after splitting the log term as a sum of three log terms

[static oak]

sorry I have no idea what you mean by splitting it into 3

We have to end with 2

[south dirge]

‘

[static oak]

Yea I know how to do that with 2 not 3

[south dirge]

oh its the same with any number of terms

log(abcdef…) = log(a) + log(b) + log(c) + log(d) + …

[static oak]

I also don’t know how to know what to add it to make 30 there are to many options

[south dirge]

yeah which is why we only use the favourable options

[static oak]

Sorry I’m just very confused I was out for the lesson before this and my teacher was out for this lesson

[south dirge]

okay lets start over again

[south dirge]

Was this clear tho?

no?

[static oak]

not really

[south dirge]

okay thats fine

[static oak]

Cus I just don’t undestand how I’m supposed to know what to add with

[south dirge]

What abt this

[static oak]

Cus u have 29+1,28+2, etc

That made sense

[south dirge]

That wrong!!

abcd indicates product

We can only split the product!

[supple pawn]

When you add logs with same base you mutiply

[static oak]

I’m so confused

I did split 30?

[south dirge]

okay what did you do in the prev example?

as in what did you do to the 24

[static oak]

I think I copied the first part of the review guide I don’t remeber it was on Friday

[south dirge]

[static oak]

I just don’t know what’s going on cus I missed class

and I don’t understand it at all

[south dirge]

okay uh, we need terms like xlog(2) , ylog(3) right

[static oak]

yea

And I know 3•10 is 30 but 2 can’t become 10 so I can’t do that

[south dirge]

wait hold it there!

Good job

[static oak]

6•5 can become 30 but 3 can’t become 5

[south dirge]

can you do this step first?

[static oak]

But that doesn’t work

[south dirge]

yeah ik, but we are making progress

[static oak]

But I can’t do it as the first step since it doesn’t equal that so it’s untrue

[south dirge]

?

why not

[south dirge]

Use this, keep a as 3 and b as 10

Is this step clear?

[static oak]

Sorry I’m confused

[south dirge]

And yes yes ik we have a 10 but, still we got a log3 term

[static oak]

But 2 can’t become 10 with ^2 or ^3

[static oak]

Don’t we need a log2 term not log 3?

[south dirge]

Thats fine, you forgot about the thing that log_5 (5) = 1

[static oak]

What…

[south dirge]

so what i mean is that, log_5 (10) = log_5 ( 2•5) = log_5 ( 2) + log_5 (5)

[static oak]

I’m so lost

[south dirge]

Okay observe this for now, lemme know if this was straight

[static oak]

It makes sense but I don’t understand how log5(5) is 1

[south dirge]

Well its the definition of logarithm

log _a (a) = 1

[royal gust]

5^1 = 5
So log5(5) = 1

[south dirge]

It may feel like a lot, but just know that its all a play based on logarithm formulae

Have a list of them by your side while you solve future Qsns

[static oak]

Idk I just feel screwed for the rest cus I feel like it makes sense when people explain it after a while but I can’t do it myself

[south dirge]

you can find the list anywhere on web by js searching it up

[south dirge]

yeah i feel you, which is why it would be better to have a list of those formulae with you

dw, log is extremely simple

[static oak]

I feel like I may even get one on the test I got smth similar for the last unit cus on my iep I get a small reference sheet on tests. Tho most of them are useless besides lasts units

[royal gust]

Honestly, I get a lot of mileage out of "taking the log of both sides" or "taking the exponent of both sides"

[static oak]

wdym by taking the log of both sides? Like change of bases?

[royal gust]

log5(5) = 1
Take the 5^ of both sides:
5^log5(5) = 5^1
5 = 5^1

That's a way to go from "log view" to "exponent view" without having to memorize how to rearrange

[static oak]

Isn’t that just rewriting the log

[royal gust]

Yeah that's exactly that formula, but I got from (1) to (2) by doing a 5^ on both sides

Maybe useful to you, maybe not. I find that can be useful often

[static oak]

Don’t I have to get rid of the extra 1?

[late gazelle]

$\log_a({a})=1$

[soft zealot]

KB

[royal gust]

Oh no you don't have to get rid of that

You can't get rid of that

[static oak]

But I have to rewrite things only in a and b so idk what to do 😅

[royal gust]

I think adding 1 is acceptable. An unacceptable answer would include a log

[static oak]

kk

For this one sorry I can’t out the question it’s simplifying smth I understand how to do it but I don’t know how they got rid of the log3(2)

[royal gust]

Change of base formula

Where c = 3,
b = 16, a = 2

[static oak]

Ohh ok Ty

[final saddle]

@static oak Has your question been resolved?

[final saddle]

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[versed cargo]

is there any way to solve this equation without using the Weierstrass substitution?

[rough crystal]

Hmmm, couldn't you do sin^2(x) + cos^(2)(x) = 1, solve for either sin(x) or cos(x) and then sub that in and solve it? perhaps treat sin(x) as y and cos(x) as x (unit circle), just for purpose of substitution until you have done the algebra and reach an and result then plug them in?

[onyx peak]

the left hand side can also be rewritten as Rsin(x+a)

[versed cargo]

wouldn't that yield a ± which could be troublesome? also i didn't understand the second part of your message

[versed cargo]

in polar form?

[night raft]

no harmonic form

[onyx peak]

Rsin(x+a) = Rsin(x)cos(a) + Rsin(a)cos(x)

so if you can find R and a s.t.
Rcos(a) = 3
Rsin(a) = 4

then youre done

[versed cargo]

im unfamiliar with harmonic forms. perhaps you could teach me something about it?

[opaque nexus]

Divide both sides by 5

[final saddle]

@versed cargo Has your question been resolved?

[onyx peak]

https://www.desmos.com/calculator/ah2kgidvmp

if we take a circle of radius 3 and a circle of radius 4, these 2 lengths give us sin(x) and cos(x)

notice that the orange circle is rotated 90° compared to the red one, thats the phase shift between sin and cos

[onyx peak]

is it clear why those 2 lengths are 3sin(x) and 4cos(x)? x is this angle

[versed cargo]

yes, sinx would be y-coord, and cosx is x-coord but you said it's rotated 90 deg just like in unit circle

[onyx peak]

yep, great

https://www.desmos.com/calculator/uwodzasvhl

is the animation in this graph playing for u?

[versed cargo]

i could run it myself, but sure

what does the largest circle show?

[onyx peak]

the largest circle shows adding those 2 movements

if you add this red and orange rotating stick, they will rotate in a bigger circle - the purple one

and the sin on that bigger circle will be blue + green, 3sin(x) + 4cos(x)

[versed cargo]

and then find x s. t. blue + green would equal to 3 (or red stick in our case)?

[onyx peak]

Yeah, sure, that would work

but for our purposes, lets solve it by instead finding what that purple circle is

that way we will be able to describe 3sin(x) + 4cos(x) with a single sine coming from that purple circle

if we look at x = 0, notice that the purple radius can be computed by pythagoras

red = 3, blue = 4

[versed cargo]

yep that's 5

[onyx peak]

damn my labels are wrong on the purple circle, should be the opposite

perfect

and there is also some angle shift

this angle

[versed cargo]

dw i'll change them myself

[onyx peak]

and whats this angle btw?

youll have to use some trig

[versed cargo]

arcsin(4/3)?

[onyx peak]

almost

not arcsin

its opposite / adjacent = 4/3

[versed cargo]

ohh

arctan

[onyx peak]

Great, so
3sin(x) + 4cos(x) = 5 * sin(x + arctan(4/3))

ofc its a bit difficult to do it geometrically every time, so ill show you the algebraic way to do it

it works whenever you're adding multiply sines / cosines with same period but different phase shift

Firstly, from the geometrical insight we know that it can be rewritten as a single sine like this:
3sin(x) + 4cos(x) = Rsin(x + a)

now we use the identity Rsin(x+a) = Rsin(x)cos(a) + Rsin(a)cos(x)

3sin(x) + 4cos(x) = Rcos(a) * sin(x) + Rsin(a) * cos(x)

now we can match the coefficients of sinx and cosx

3 = Rcos(a)
4 = Rsin(a)

and now we need to solve this system

we can divide those equalities and get
Rsin(a) / (Rcos(a)) = 4/3

sin(a) / cos(a) = 4/3

a = arctan(4/3)

and we can square them and add them

9 = R^2 cos^2(a)
16 = R^2 sin^2(a)

25 = R^2 (cos^2(a) + sin^2(a)) = R^2 * 1
R = 5

so this is the algebraic way to solve it

[versed cargo]

i see

[onyx peak]

and once you rewrite it with a single sin, its pretty simple to solve

it becomes a fairly standard trig equation

[versed cargo]

and then sub these value back into here?

[onyx peak]

yep

and we get that it equals 5sin(x + arctan(4/3))

so the equation simplifies to
5sin(x + arctan(4/3)) = 3

and this can be solved by normal trig

[versed cargo]

so

i got

x = (-1)^k*arcsin3/5-arctan4/3+kpi

where k belongs to Z

[onyx peak]

looks correct

[versed cargo]

yippee

thx so much

[onyx peak]

np

[final saddle]

@versed cargo Has your question been resolved?

Channel closed

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[manic leaf]

\subsection{Problem 6.} \textbf{Suppose that $(X, \leq)$ is a partial order. A subset $A \subseteq X$ is \textit{pairwise incomparable} if, for all distinct $a, b \in A$, $a \not\leq b$ and $b \not\leq a$. A pairwise incomparable subset $A\subseteq X$ is \textit{maximal} if there is no pairwise incomparable subset $A' \supseteq A$ with $A' \neq A$. Choose your favorite equivalent of AC (i.e., (ZL), (HMP), or (WO)) to prove that every partial
order $(X, \leq)$ has a maximal pairwise incomparable subset}

[soft zealot]

toast

[manic leaf]

hi

If X denotes a linear order

Then what would a maximal incomparable subset look like? Just a singleton?

[onyx peak]

Yes

[manic leaf]

Alright

I am not sure if I am stupid

Would the idea be like

Find chains of X

Then take the complement?

I have to think about it a bit more

[onyx peak]

if you take a complement of a chain, it wont necessarily be pairwise incomparable

take a poset like this, then the middle chain's complement is just the other 2 chains, and some of the elements of those are comparable

[manic leaf]

Ur right

[onyx peak]

Personally, i'd try to use ||zorn's lemma|| here

[manic leaf]

hmm

I have an idea

What if I take the set of maximum of every chain

wait

oki this doesn’t work because there can be a lot of chains

[onyx peak]

it would work, if the chains had a maximum

[manic leaf]

oh

[onyx peak]

but they dont necessarily have it

[manic leaf]

so this is where zones lemmas come in

[onyx peak]

for example the Z chain doesnt have a maximum

[manic leaf]

What jf we consider chains with an upper bound

[onyx peak]

what if there are no such chains?

for example take a poset that consists of many Z chains

[manic leaf]

hmm

[onyx peak]

Can you just restate what zorns lemma says?

[manic leaf]

If X is non empty and if a every chain has an upper bound in X, then X has a maximal element

[onyx peak]

yeah, so zorns lemma allows you to prove existence of maximal element, under some conditions

[onyx peak]

this problem asks you to prove existence of a certian maximal thing

my hint is to focus on the goal and not on the poset you have, zorn's lemma wont be applied directly on the poset you have

[manic leaf]

Hmm could I consider a set of maximal pairwise incomparable sets

Maybe order them by length

[onyx peak]

a small issue is that it could be empty (the fact that its nonempty is what youre trying to prove afterall), and the length isnt necessarily a natural number, so idk how would you define that

but you're not that far

[manic leaf]

Ah

[onyx peak]

ill let you work on it and if you want another hint, lmk

[manic leaf]

Kk

[manic leaf]

ok ngl

Is it possible for a set to look like this

[onyx peak]

to look like what?

[manic leaf]

Infinitely big but none of the terms are comparable

[onyx peak]

yeah, a poset could look like this

the order relation would just be empty

[manic leaf]

What would be the maximal incomparable subset?

Oh

[onyx peak]

the set itself

[manic leaf]

It would just be the set itself

Ic

So that’s why we can’t rlly use length easily

I want to be able to generate a set

With incomparable elements

Until I cannot anymore

[onyx peak]

Yeah, you want the maximal incomparable set wrt to the subset ordering

[manic leaf]

Ah

oh man that is so smart

[onyx peak]

do you get what im hinting at?

[manic leaf]

bc that works for infinite set

cuz the biggest subset is the set itself

so you never have to worry about avoiding an upper bound

[onyx peak]

Well, I essentially just re-read the problem

they ask you for a maximal pairwise incomparable subset and they tell you wrt what ordering

do you know to what poset you should apply ZL now?

[manic leaf]

wait I am a little confused about subset ordering though

the blue circle and green circle are both examples of incomparable sets right

[onyx peak]

yes

[manic leaf]

Is it true that the blue set is a subset of the green one?

[onyx peak]

no

and the converse isnt true either

neither is a subset of the other one

because they contain completely different points

[manic leaf]

ah I guess they don’t need to be subsets of each other then

[onyx peak]

indeed they dont. Zorns lemma works on any poset satisfying the conditions, not just on total orders (you dont even need AC for those)

[manic leaf]

Oooh okay I see

Cuz zorns lemma says nothing about comparability

Every chain has an upper bound means in this case it’s just the “widest” extent of something

[onyx peak]

Zorns lemma just needs it to be a poset, and all elements of posets arent necessarily comparable

[manic leaf]

This widest extent will always be bounded because at most it can contain elements of the original set itself

[onyx peak]

oh btw im not sure whether we're on the same page already

[onyx peak]

do you already know what poset with what ordering you should apply zorn's lemma to?

[manic leaf]

I think we might be, I think we consider the set of incomparable subsets of X with respects to subset ordering

[onyx peak]

exactly, because you want the maximal element of that poset, you're essentially forced to apply zorns lemma on that precise poset

[manic leaf]

the biggest possible comparable subset of X wrt to subset ordering is X itself

[onyx peak]

comparable or incomparable?

[manic leaf]

Incomparable*

[onyx peak]

that was only the case in here

[manic leaf]

Yep

In most cases it’ll be less

[onyx peak]

yeah

[manic leaf]

but regardless it’ll always be upper bounded

[onyx peak]

why?

if you want to upper bound it by X itself, then you cant do that

[manic leaf]

hmmm

[onyx peak]

because X isnt necessarily an incomparable subset of X, its not even part of your poset (the one of incomparable subsets ordered by subset relation)

[manic leaf]

consider we have some set

uh

we start by adding incomparable terms to this set

every time we generate a new chain

If we cannot add a new incomparable term

We are done

And we found an upper bound

Otherwise we keep adding elements until we reach X itself

[onyx peak]

if this worked, you wouldnt even need zorns lemma, because once you cannot add new incomparable term, you've found a maximal incomparable subset.

[manic leaf]

Hmm

[onyx peak]

you're describing some vague possibly infinite process (or even transfinite one could say, since even after e.g. countably many steps, you still wouldnt necessarily be done)

[manic leaf]

Maybe I need to select an arbitrary chain in our new posey wrt subset

And show it must have an upper bound

[onyx peak]

If X is non empty and if a every chain has an upper bound in X, then X has a maximal element
have you already shown that its nonempty btw

[manic leaf]

I said this

Oh wait

Show X is no empty right