precalculus

[willow skiff]

so it's similar here, if you understand this concept
that you can mess up the translation you want if you don't follow the order
then you're good to go, if you practice of course

[unborn brook]

Sure but my point is at some point we have to define arbitrary stuff to avoid ambiguity

[willow skiff]

sure

in terms of mathematical notation and communication, absolutely

the issue here is your problem is about understanding how transformations work, right?

[unborn brook]

it's not that I'll memorize some distorted mnemonic to deal with the operations

I just wondered why it is the way it is

[willow skiff]

honestly I think this is doing you more harm than good

the picture

[unborn brook]

I'd say it's about the undercurrents of the convention

[unborn brook]

What do you mean

[willow skiff]

I wouldn't memorise this order

cause it's not even true that you have to follow that order to get it right

[unborn brook]

Yup I've realized that translations and transformations are inherently interchangeable

[willow skiff]

cool

[summer ruin]

there are operations for which order doesn't matter and those for which it does matter

[scenic coral]

which is why division sucks

[terse aspen]

Hello can someone help me on describing the end behavior and any ezcluded values with limit notation in dms pls

[winter comet]

do you have a public question 👀

[terse aspen]

Hi yes

Sure

So i have this

Y= 4x/x^2-4

Im doing the top problem

[winter comet]

y = 4x/(x^2-4)?

ok

doesnt send full problem 😔

[terse aspen]

Domain and range is easy but im confused on “describing the end behavior and any excluded values with limit notation”

[terse aspen]

^

[winter comet]

did you come up with domain restrictions?

two of them :>

[terse aspen]

Yes

[winter comet]

do you know limits?

[terse aspen]

Domain is all reals and x cant equal plus/minus 2 and y is all reals

[winter comet]

yeah

[terse aspen]

Yea i know basic limits and stuff but i forgot more complex since this a 3 way thingy graph

[winter comet]

do you know the limit approaching these domain restrictions/excluded values of x?

[terse aspen]

Yes

Plus or minus 2

[winter comet]

well you'd look at the limit as x approaches -2 or 2 right?

[terse aspen]

So theres 6 total lim?

Yes

Ok wait let me do and see if im right?

[winter comet]

4

approaching 2, approaching -2, approaching infinity, approaching -infinity

those are your limits, right?

[terse aspen]

But what about middle

[winter comet]

what about the middle?

[terse aspen]

Yes

[winter comet]

is there a discontinuity there XD

[terse aspen]

Isnt it somthing like

[winter comet]

is 0 an excluded value?

[terse aspen]

-infinity

No

[winter comet]

then why are you looking at the behavoir approaching 0?

[terse aspen]

Idk tbh

[winter comet]

it tells you to look at end behavior (as x approaches +-infinity) and excluded values (as x approaches +-2)

so you'd have four limits

[terse aspen]

Oh ok

So would there be any case where we are looking at 6 limits

[terse aspen]

Mot end behvauor of all only excluded values sections?

[winter comet]

with the same instructions, the function would have to have four discontinuities like this instead of two ^

[winter comet]

end behavior just means approaching infinity or -infinity

thats seperate from the excluded values

the end behavior is 2 limits

always

but in this case there are two excluded values and one limit for each value

[terse aspen]

Oh ok

I see ok

So only 4

Can you help me qith another problme

[winter comet]

what is the other problem?

[winter comet]

what is the denominator?

x^t 8x^6 - 12x^3?

[terse aspen]

X +8x^6 -12x^3

[winter comet]

oh LOL

[terse aspen]

My bad lmao

No t

[winter comet]

whats the highest power term in the limit?

or just the highest power in general lol

[terse aspen]

6

[winter comet]

yeah

[terse aspen]

In the denominator

[winter comet]

try dividing the numerator and denominator by x^6

x^6 sorry

💀

[terse aspen]

Wait what

:/

[winter comet]

divide the numerator and denominator by x^6

so like

[(13x^3 - 1502x^2)/x^6]/[(x+8x^6 - 12x^3)/x^6]

you see what i mean?

[terse aspen]

Like long division?

Yes

[winter comet]

no

like (63x^3 - 1502x^2)/x^6 = 63/x^3 - 1502/x^4

right?

[terse aspen]

Wait

The top is 63

My bad

[winter comet]

oh

63 then

same process XD

[winter comet]

i cant read 💀

[terse aspen]

All good

Why are we doing that

[winter comet]

same answer in fact 💀

[terse aspen]

I have that now

[winter comet]

so then the denominator becomes

(1/x^5 + 8 - 12/x^3) right?

[terse aspen]

Wait one sec

[winter comet]

so the limit should become this

[terse aspen]

Ok i havd that

[winter comet]

ok

now as x approaches infinity, what happens to 63/x^3?

[terse aspen]

Smaller

[winter comet]

right its approaching 0

[terse aspen]

Yes

[winter comet]

same with any term with x underneath

so lim x->infinity 0/8 = 0, right?

[terse aspen]

Yes only constant in the bototm

Yes

[winter comet]

yea

[terse aspen]

So equal 0

[winter comet]

so the limit is just 0

[terse aspen]

🤯

[winter comet]

fun fact: you can just take the term with the highest power in the numerator, and the term with the highest power in the denominator, and take the limit of the highest power in numerator divided by the highest power indenominator

so like, here it would be lim x->infinity 63x^3/8x^6 = lim x->infinity 63/8x^3 = 0 still

[gritty totem]

there are rules regarding limits of infinity in the same way that there are rules when finding asymptotes of a rational function

[winter comet]

because all the other terms are going to 0

[winter comet]

yeah...?

what about it 💀

[gritty totem]

oh no, i was just saying because i saw the question and thought of a faster way of doing it

[winter comet]

oh you probably mean that ^

or something liek it lol

[gritty totem]

yeah

[terse aspen]

Omg

I did not know this

[winter comet]

well its a faster way but hopefully you see why it works

it only works with infinite limits tho

[terse aspen]

I am catching up on 5 pre calc lessons and 2 bc lessons

[winter comet]

(or approaching -infinity)

[terse aspen]

Can i have help with 1 mire

Laast one

[gritty totem]

mind if I help?

[terse aspen]

Yea sure

So i have y=3x/ absolute value phrase x-1

And i have to find the domain and range of the function describe the end behavior and any excluded x values with limit notation

[gritty totem]

the absolute value of x-1 is equal to two functions:

-(x-1) if x < 1 and (x-1) if x> 1

the domain is, of course, how far to the left and how far to the right it can go

[terse aspen]

Yes

I see how -(x-1) works but how does the negative get there

[gritty totem]

it has a vertical asymptote at x= 1 as well

[terse aspen]

And horizontal of 3 ??

[gritty totem]

positive and negative 3

that’s why the -(x-1) is important in this case

[terse aspen]

But than what about this graph

Thats the graphbof the equation

How does the horizontal work here

[gritty totem]

horizontal asymptotes are positive and negative 3

[terse aspen]

Oh wait i see that

Yes

[terse aspen]

But how? Than

Is it allowed to go through?

[gritty totem]

-(x-1) = 0
becomes x - 1 = 0
x = 1

x-1 = 0
x=1

[terse aspen]

Ok

I see

So whats next

[gritty totem]

you’re also looking for the domain and range

the domain is how far left to right the function can go

the range is how low to high the function can go

[terse aspen]

So x all real and y > -3

[gritty totem]

let me see:

domain: (-infinity to 1) U (1 to infinity)
i’m not sure about range though

[terse aspen]

Hole ok

So the next part is describe the end behavior and any excluded X values with limit notation

[gritty totem]

oh i see

[gritty totem]

we can say that the limit of the function as it approaches 1 is infinity

[terse aspen]

And thats it

Oh ok wow thats weird

What about the other 2

As it aproaches infinity y is infinity and is it aproaches negative infinity y is negative infinity?

[gritty totem]

limit as the function approaches +- infinity is -3 and 3

[terse aspen]

Oh my god this is making sense

I know i got it wrong originally but still

Tha k you

[gritty totem]

i’ll give you proof if you want to look into detail

or i can explain it to you

whichever option

[terse aspen]

Wdym

[gritty totem]

as to why the limits approach infinity make sense

[terse aspen]

Yes pls

Explain

[gritty totem]

so, this is where the rules of a rational function come into play regarding limits approaching infinity

one of the rules is that if the power of the numerator is equal to the power of the denominator, then the horizontal asymptotes is equal to the coefficient of the numerator divided by the coefficient of the denominator

here, with the function that we were solving, the powers of both the numerator and denominator were equal

that’s how we got 3 and -3

so, with limits of infinity, the rule i mentioned still applies since the powers of both the numerator and denominator are the same

[terse aspen]

Oh ok yea and what gets the denominator to 0 is vertical asympte? But how does slant higher top than bottom or higher bottom than top power wise

[gritty totem]

i do need a little clarification on what you’re trying to ask me with this question

also, yes, the vertical asymptote is obtained when you set the denominator of the function equal to zero

[terse aspen]

How would i get the f(x) nunber of the limit if it is set to inifinity but the either the numerator power is higher than the denominator or vice versa

[winged peak]

Hey guys, so Im learning functioms amd polynomials in my precalc class and I am so lost on the graphimg padt

So, I thought since the degree was positive, the graph falls to the left and since the leading coefficient was positive, it rises to the right

I failed a question bc I got the "a" part wrong

Could anyone help me explain how to understand the "finding the end behavior" better?

[gritty totem]

oh when the power of numerator is greater than or less than the power of the denominator?

greater than: the horizontal asymptote does not exist

less than: the horizontal asymptote is 0

[terse aspen]

Yes

Oh ok

I see thank yiu fro all the help

[gritty totem]

also, this means that limit approaching infinity is infinite for “greater than” and 0 for “less than”

[winged peak]

Is anyone gon help me... halp

[gritty totem]

i was reading your question and i’m here to help

[gritty totem]

i just needed to help Vibez with the final part of their question

[winged peak]

Thank you, thank you thank you SMMMMM

Ok this ia gon take me an hour to despipher ill brb 😭

[winged peak]

If u can help explain in simpler terms id appreciate

[gritty totem]

@winged peak i do want to add with the function behavior:

for example, when they tell you as x approaches infinity -> f(x) approaches infinity, it’s telling you to read from left and right or right to left

[winged peak]

Just in advance, I have a slight lesrning disability so im kinda slow 😭

[gritty totem]

we’ll still help regardless

[winged peak]

Thank you, Im new to my university and i have no intention of fucking around over here 😭

[terse aspen]

If even highest degree or combined degree they go same direction if lead coefficient is greater than 0 than it goes up both sides if less than 0 they go down both sides

If odd highest or combined degree they go opposite side

If greater than it goes down than up if less than 0 it goes up than down

[winged peak]

Alr I uave to do a DIFFERENT question thats not the one i showed on paper, but ill ask you for help later. Its ok if u dont anseer immediately or jave othee things

[terse aspen]

Draw your arrows than your bounces sqiuggles and straights if your graphing

[winged peak]

Yes im graphing on a computer i just showed my work on paper

[terse aspen]

Am i right @gritty totem ?

[gritty totem]

yeah

and you can factor the function to see where the zeros are located and use the information to sketch the graph after

[terse aspen]

Yep but if its factored out dont take the highest degree there you need to add them up including if there is a x without a clearly defined power its 1 if not another

[gritty totem]

yep

[terse aspen]

Can i dm you like monday or something since i got a test Wednesday

[gritty totem]

sure

and what’s the test about?

[terse aspen]

The bc 1.1 is easy its just areas of rectangles for the underside of a quadratic thingy

[gritty totem]

oh ok

[cerulean spire]

hi, im new can anyone pls explain what calculus is. I really would like to learn

and teach me the basics and stuff

[gritty totem]

calculus is the study of rate of change and it’s split into two categories: differential and integral calculus

differential calculus is finding the rate of change of a function such as derivatives (which is the slope of a function)

[winter comet]

he doesnt know algebra either 😔

[gritty totem]

who? thecoffeedrink?

[winter comet]

yeah

#prealg-and-algebra

💀

[gritty totem]

oh i see

i don’t mind someone being curious to calculus at all, but if they’re wondering what algebra is, then algebraic work has to be covered first

[winter comet]

yeah xD

[winter comet]

i'm not sure if the explanation would work without knowing algebra XD

could just say "study of changing things and....infinite things :>"

😂

[gritty totem]

oh i see

[winged peak]

Ayo could u fact check this

Nvm it was right I was being delusional

[gritty totem]

This is correct

[winged peak]

Bro, the website I do my hw on is called aleks

and if I get 1 problem wrong, I'm assigned more questions

And it's like I'm traumitized at this point and in tears

[gritty totem]

That name hasn’t popped up in my mind in years

[winged peak]

Fr?

R u in a university?

[gritty totem]

Yep

College freshman

[winged peak]

This website is fucking ass like it charges me money to do my damn hw and Ion even use the damn e book

[winged peak]

SLAYYY MEE TOOO

[gritty totem]

I have multivariable calculus and intro to proofs - which is why I have experience with precalculus

[winter comet]

i have experience with precalculus but i never took it

[gritty totem]

and my homework is mainly on paper; some is through technology but it’s for a “quick check” from the concept i read for intro to proofs

[winter comet]

it still technically counts though...

[winged peak]

Wait

this channel banns the topic regarding ... " i - q"

Bc I was literally tryna compliment u omg 😭

[gritty totem]

what exactly?

[winter comet]

i technically still have experience even though i didnt take it

i mean like idk if it can be called 'experience' but

:l

[gritty totem]

oh i see

[winter comet]

💀

[gritty totem]

but hey, there’s nothing wrong with self-teaching yourself a subject

[gritty totem]

they do?

[winged peak]

Yeah

I just responded to u "Wow, ur i - q must be up there w albert" and the server banned it due to irrational arguments over it idk

[winter comet]

try it

[winged peak]

Honestly Ig for a server w 200k users, sounds reasonable

[gritty totem]

i guess as well

[winged peak]

Idk how long u been here but has this disc chat had any chaotic moments

[gritty totem]

just joined today so i don’t have much of a clue

[winter comet]

usually pretty chill 💀

[winged peak]

Thank god. Hope yall dont mind my username I would change it but this server doesnt allow LMFAO

[winged peak]

Oh, and ur one of the smart ones too! Please verify if this is correct. If I get this right, this is the last question I have to do.

[winter comet]

😂

[gritty totem]

Yep, this one’s right

[winged peak]

Alr tysm for clarifying bestie. Transitioning to college has been like, kinda wild if I'm being honest. I love it, but it's quite difficult to adjust here and there. The math hw I was just doing was only like, prep, which was tedious asf.

And the weird part is, the math prep usually has nothing to do w the exams, or lecture. Like I barely passed my first exam which is giving me hella anxiety. 😭

It's nice to meet u tho! Hope to talk to u more < 3

[gritty totem]

i would say the same as well in terms of transitioning because i have a more lighter schedule for classes whose difficulty increases as the semester passes.

and with math, the more practice you do, the better; i do believe by your next semester, you take calculus - i could be wrong but i’m just predicting in the case of it all

[gritty totem]

likewise

[winged peak]

Yeah that's correct im moving to.. actual calculus? idk if I'm dying here, it's probably bc of the professor smfh

Do u dorm ???

[gritty totem]

no, my home is near my university

[winged peak]

ANYWAYS I GTG TO WALMART AND BUY TOILET PAPER BYEEEE

[winged peak]

Oh, slayyyy. I dorm bc I can't drive

[winter comet]

toilet paper is such a life saver ngl

[gritty totem]

oh i see

[winged peak]

Fr bc all I eat in college all day is Panda Express

[winter comet]

yum :>

[safe basin]

Bros name

[daring tapir]

😭 dude why tf did i read this

[stiff sluice]

Wait

[daring tapir]

sub 1/x = t

[safe basin]

af(bx-c)+d vertically stretches f by a, horizontally stretches f by b, horizontally translates f by c, and vertically translates f by d

Obv each of those can be done individually too or in whatever combinations but I just stuck it all in one place for conciseness

So for these to be correct you’d have to stick the vertical translation outside of the argument of f

Cause f(x) + k vertically translates f by k

For the factoring thing it depends, you just want to be able to express it in a form where you can extract this info relative to some function f

[earnest flower]

Can someone check if these are correct

g(x)=2x^2 -6x+4
h(x)=-2x^2 +10x -12
j(x)=3x^2 -12x +9

[willow skiff]

,w polynomial through (-3, 40), (1, 0), (2, 0)

[obsidian monolith]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by sooouth.

[willow skiff]

,w polynomial through (-3, 0), (1, 8), (2, 0)

[obsidian monolith]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by sooouth.

[willow skiff]

,w polynomial through (-3, 0), (1, 0), (2, 15)

[obsidian monolith]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by sooouth.

[willow skiff]

check your answers yourself

only g(x) is correct

[topaz wave]

do you have reviewer on pre cal guys?

[willow skiff]

https://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx

[earnest flower]

i dont understand how to find this

Nvm

I didnt know if the - - 3 would make it positive when factoring now i jnow

[exotic barn]

I sent a video in your help channel and you deleted your msg, you're weird for that

[rotund jasper]

sorry if it's wrong channel but can u guys help me this

Given sequence a1= 1, a1<a2<...<ak such that these positive integers < m and coprime with m. Prove that if the sequence a1,a2,...am is an arithmetic progression with at least 3 numbers, so m will be a prime or an exponential of 2

[cloud escarp]

hello guys
Im new here can anyone suggest me or guide me where i should start from.
Im in High School

[exotic barn]

you might wanna check out Professor Leonard

on YouTube

[cloud escarp]

thanks for the suggestion

anything else would u like to suggst for me

[exotic barn]

maybe start

[daring tapir]

😭

[keen swift]

I am nerd

[warm cove]

Precalculus test was easier than I expected, I am definitely improving.

[slim lark]

Hello everyone I wanted to start calculus but I lack many concepts I have studied math till 10th grade Real Numbers
Polynomials
Pair of Linear Equations in Two Variables
Quadratic Equations
Arithmetic Progressions
Triangles
Coordinate Geometry
Introduction to Trigonometry
Some Applications of Trigonometry Circles
Constructions
Areas Related to Circles
Surface Areas and Volumes Statistics
Probability. So I wanted to know what else do I need to cover

can anyone help

[restive glacier]

Yes here.

[slim lark]

someone recommend me to cover functions, integration, differntiality,limits and continuity

is it enough?

[slim lark]

.

[exotic barn]

Professor Leonard

[slim lark]

I know but

[restive glacier]

Yes. Enough to ability.

[slim lark]

ok thanks

highschool

[restive glacier]

OK. high school

[slim lark]

ya what about you?

[restive glacier]

I graduated from college.

[restive glacier]

Your age?

[slim lark]

17

[restive glacier]

Ok. good.

I am 26

[slim lark]

what major?

[sleek folio]

How do I proceed to prove this?

[river drift]

,rccw

[low yoke]

Hey there! can someone explain me the difference between the slope and the rate of change? I understand them as the same, am I correct?

[winter comet]

yeah, they're the same

[low yoke]

And why they got different name?

[winter comet]

why do you call them differently you mean?

💀

that was supposed to be a joke

[winter comet]

anyways one thing can have multiple names, its how english works ☠️

[low yoke]

I meaned, why In my book it has like 2 different terms for the same stuff

im not native english :/

[sleek folio]

Slope is defined for straight line

Rate of change is for any function

[river drift]

"slope" is an interpretation of the derivative driven by the graph of a function, whereas "rate of change" is driven by interpreting a function as modeling a physical quantity

[low yoke]

Okay, understood

thank you guys

❤️

[serene crag]

i just got a 43 on my precalc test

am i cooking

[upbeat yarrow]

no

[winter comet]

cooking at mcdonalds maybe

☠️ ☠️

😂

[serene crag]

idk what i got wrong either

i just have a non-curving ap teacher

[winter comet]

😔

[winter comet]

unlucky

[serene crag]

😭

she doesnt even follow ap cirriculum

she makes her own

[winter comet]

😭

[terse aspen]

Can someone help me on piece wise functions they are easy I just dont grt this problem 😐

[winter comet]

whats the problem

[terse aspen]

Number 3

One of them is dne I think its the first but idk why

[winter comet]

have you tried graphing it? it might help with understanding whats going on

[terse aspen]

I have not but on my test there wont be enough time he said

[winter comet]

but for (a) you're gonna wanna look at the graph approaching 2 from the right and from the left

[winter comet]

well you see that at x=2 the graph changes elementary functions, right?

because its -x^2 + 5 for x<2 (on the left side of x=2) and 4x for 2<=x<4 (on the right side of x=2 including x=2)

so you're going to want to know lim -> 2- (-x^2 + 5) and lim x->2+ (4x)

if they evaluate to the same thing, then the limit is what they both evaluate to

if they evaluate to different things, the limit does not exist

am i making sense or nah 💀

[terse aspen]

…

[winter comet]

💀

[winter comet]

you might want to graph it this time just for a bit of visual representation

[terse aspen]

Bro idek how we got to evaluting them

Ok

One sec

[winter comet]

yeah

well first of all, as x approaches 2 do you see graphically that the limit doesn't exist?

like both sides approach different things

[terse aspen]

Yes

Ita like this one

[winter comet]

lim x-> 2- f(x) is not equal to lim x-> 2+ f(x) right

[winter comet]

yeah exactly

but we can find numerically where each side approaches

[terse aspen]

Isnt it becuase 1 value having more than 1 y value

[winter comet]

without graphing it

[terse aspen]

How

[winter comet]

do you know about one sided limits?

[terse aspen]

Yes

[winter comet]

so lets say you're approaching f(x) from the left

lim x-> 2- f(x)

since you're looking at x values less than 2, which part of the piecewise function can you use?

remember the function is -x^2 + 5 if x<2, and 4x if 2<= x < 4

[terse aspen]

X<2

First one

[winter comet]

yeah so you're gonna use the function for x<2 right

so as x approaches 2 from the left, lim x->2- (-x^2 + 5) = the value you're approaching from the left

make sense?

[terse aspen]

Wait are you literally subtracting like this or a dash

[winter comet]

lim x-> 2-
the - after 2 means its approaching 2 from the left

lim x-> 2+ would mean approaching 2 from the right

i shoulda explained that before 💀

[terse aspen]

Yea i know that part

When i look at it

Just dont know how to see this qithout a graoh just mind

🧠

[winter comet]

like you can just think of some arbitrary graph but you only care about the graph for x<2

[terse aspen]

Ok

[winter comet]

then approaching from the right, it should be the same value if the limit exists, right?

[terse aspen]

Yes

[winter comet]

so lim x->2+ f(x) is the limit of f(x) as x approaches 2 from the right

which part of f(x) do you use?

you're only going to realistically use the values to the right of 2 that are close to 2

[terse aspen]

Yes

[winter comet]

remember f(x) = {(-x^2 + 5) if x<2, 4x if 2<=x<4, and x^2 if 4<= x
if you're approaching 2 from the right but you only care about values close to 2, which of these do you use?

[terse aspen]

Only option is 2<_ x <4

[winter comet]

right

so you're going to use the 4x part of the function

so lim x->2+ f(x) = lim x-> 2+ (4x) = the value of the limit as f(x) approaches 2 from the right

right?

🤣

[terse aspen]

No

The y values do not equal each other

Wait what

[winter comet]

the one sided limits aren't equal yeah

[terse aspen]

Ok i see

[winter comet]

basically you have to determine from which way you're approaching, and use the appropriate part of the piecewise function. if the limits from both sides both evaluate to the same number, then the entire limit is that number. But if the limits from both sides evaluate to different numbers, the limit doesn't exist

[terse aspen]

Oh i understand now

Thank you so much

🤯🤯

[winter comet]

😂

[terse aspen]

How do i know thag my line parobola thingys are going thr same direction if its like this

[winter comet]

i guess you could take the limit

or just make a table of values as x approaches your discontinuities 💀

[terse aspen]

But is there not a way of just knowing that the graph will both be up vs being on opposite axes like 1 and 3 or 2 and 4

[winter comet]

you could take the derivative and see if its increasing or decreasing on the intervals 😁

but i don't know of an easy way to know, maybe there is an I just don't know it tho

🤷‍♂️

[terse aspen]

Oh ok thanks tho

[terse aspen]

To advanced for me 😓

[winter comet]

i'd just plug in values to see xD

[terse aspen]

Thank you tho

Lmao ok

[viscid thistle]

hi im in Gr 12 in pre calc with a mark of 64, its because of my first test. and right now im doing polynomial functions in pre calc. i know how it works but the word problems, applying polynomial problems in real life is a big challenge for me so i beg that someone give me a lengthy advice and some support please and thanks

[winter comet]

do you have a sample problem that you struggle with so we can see the type of problem?

[wanton parcel]

Any single guys my age group plz dm me I want a bf to show my friends in school 😭

Rules :
17 or under

[uncut mulch]

@wanton parcel Please don't use this server for that kind of stuff.

[hasty lily]

Hii I'm taken

[willow skiff]

worst year to be a minor

[warm cove]

Can my birthday come already 😭

I feel threatened

[warm cove]

I studied for the wrong thing on the no calculator part on the test

[small hedge]

Bro fuck precalc

i do not understand it

[wraith jackal]

Hello :) can someone explain to me how this works
(I also do not know what this is called)

$x^{ab}-1=(x^a-1)(x^{a(b-1)}+x^{a(b-2)}+...+x^a+1)$

[obsidian monolith]

Minute

[wraith jackal]

I know its related to (x-1) always being a factor of (x^n-1) for any natural number n but i dont get how you derive that from this

also lmk if there was a better channel to pose this question in

[safe blaze]

By induction on b

Or well by using x^n - 1 = (x-1)(1 + x + ... + x^(n-1))

[wraith jackal]

wow duh thank you idk why this didnt hit me

do you know what that theorems called

[safe blaze]

Hmm idk

It's kind of related to the geometric series formula

And cyclotomic polynomials

[wraith jackal]

ah thank you this is good reading :)

[wraith jackal]

@safe blaze apparently its the difference of powers identity with b=0 1 :P

i think this or something related was mentioned in the advanced algebra channel

yesterday 5pm aest

[tame pine]

Yo

[viscid thistle]

Yo

[warm cove]

Yo

[winter comet]

Yo

[safe basin]

Yo

[agile ginkgo]

$plot (sinx)$

[obsidian monolith]

Progno

[tame pine]

,w plot sinx

[obsidian monolith]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by oppailol.

[quasi elbow]

how do we solve question 12?

[night herald]

-1

is that the answer?

[hasty lily]

Don't just give an answer

[quasi elbow]

yes but, how did you get it?

[night herald]

yeah yeah

wait

[night herald]

i like to check my answer first

do you the sign of each trig function in each quadrant?

@quasi elbow

[quasi elbow]

i do

[night herald]

also do you know what happens when you add $n\pi$ or $\frac{(2n+1)\pi}{2}$ to the argument of trig functions

[obsidian monolith]

Black_Gold

[night herald]

??

[quasi elbow]

oh

i got it

thank you!

[night herald]

where were you stuck??

[quasi elbow]

i was writing cot(90-a) as cosec(a)

i rechecked

[fringe atlas]

https://socratic.org/questions/5a6c473511ef6b6930779a6f#542204

Can anyone explain this last part

What if the product of ac is negative?

[quasi elbow]

can someone please explain this question to me? in this, we have to find the minimum and the maximum value of the given expression

[exotic barn]

You basically have a quadratic function and you wanna find the vertex form of it

since it's open upwards, it will tell you the minimum value is 3/4

in order to find the maximum we need to consider now the vertex form expression

sin²(theta) is bounded between 0 and 1

[obsidian monolith]

bacc the sigma😔🤞

[quasi elbow]

how are we supposed to think of the process though, when we come across such questions?

if that makes sense

[hasty lily]

<@&268886789983436800> any moderators who wants a [ this person has been posting on here frequently asking for a dating partner ]

[exotic barn]

well here if you look at the terms you notice it's a quadratic in disguise

so might as well complete the square to get the vertex form etc

if it were instead of sin(x)^4 it was sin(x)^3 then it would be a bit more difficult

[quasi elbow]

got it, thank you.

[jagged drift]

@exotic barn Können Sie mir bitte mit Wahrscheinlichkeiten helfen?

[quasi elbow]

what is happening here

[viscid thistle]

where specifically ?

[quasi elbow]

um

the whole thing

[viscid thistle]

okay

so

first

this part

for some reason i feel like this is wrong

cause pi equals 3.14

but

here they are taking pi as in radians

which equals 180degress

degrees

ah

thats the reason they said this

eh

[quasi elbow]

i am still very confused

[willow skiff]

yeah that's wrong, of course pi isn't exactly equal to 3.14

[willow skiff]

but yeah sin of 1 degree is less than sin of 1 radian

sin of 1 degree is tiny

using the approximation $\sin x \approx x$ in radians

[obsidian monolith]

higher's secret twin brother

[willow skiff]

,calc pi/180

[obsidian monolith]

Result:

0.017453292519943

[willow skiff]

,calc sin(pi/180)

[obsidian monolith]

Result:

0.017452406437284

[willow skiff]

there you go, they're very close

[willow skiff]

so the graph shows you that sin is increasing in 0 < x < pi/2
and decreasing in pi/2 < x < pi

they drew a lot of lines yes

for instance, because 3 < pi, 1 < pi/3 and so sin(1) < sin(pi/3)

also sin has symmetry, so sin(pi - x) = sin(x)

so that is why sin(pi - 2pi/3) = sin(pi/3) = sin(2pi/3)

it is a lot of info yes

[quasi elbow]

i had an eight-hour study session so nothing is making sense to me anymore

i will get back to it later

but thank you

[willow skiff]

np!

[brave shard]

can anyone expalin to me how the denominator is turns into 5x-19??

[summer ruin]

multiply by x-6

[brave shard]

how does that make it 5x+19 though? shouldn't it be like x^2 -6x + 5x - 30 +4x - 24? thats what it is if you multiply x-6 with the denominator

[brave shard]

OHHH

your the goat my friend

thx

[dense eagle]

yw

[wild phoenix]

Well boys I'm cooked I'm way in over my head and there is a HIGH Likelihood I fail out of AP precalculus I will probably be here alot asking stupid questions this is a warning 🫡

[winter comet]

😔

[unborn brook]

Why is the slope of the secant line necessarily equal to the average rate of change between the two points? (I know the question is trivial and the answer might be obvious to most of you but I find this kind of counterintuitive. The slope is the same as long as the two points are the same, no matter what kind of shape the function is taking between the points. This boggles my mind. An explanation that goes "By definition..." makes it even worse.)

[exotic barn]

Say you have an interval [a,b] then you would calculate the average rate with the formula (f(b)-f(a))/(b-a) right

You wanna know how much is the average rate of change

Now if you drew a line through these two points (a,f(a)) and (b,f(b)) you would get a linear function

a line

And you would notice that the slope of that line is equal to the average rate of that function if you drew a triangle

and we call that green line a secant line because it intersects at two distinct points of f

[winter comet]

By definition, the slope of the secant line is the average rate of change between two points

jkjkjk 😂

[astral surge]

i have created brainrot of math called mathrot let me elaborate a conversation using this mathrot "wsp dude have you been integrating and graphing to make that mtan line more proportional in a equation, yea i have been how could you tell did my omega alphnul give it away or was it limits"

[winter comet]

magnificent speech 👏

[astral surge]

thank you fellow summation/sigma

[winged notch]

hi

anyone here that could help plz

[winter comet]

with?

[astral surge]

chat i was hitting some precalc when i got a message i check and this is what my friend turned into am i cooked?

[hasty lily]

I think they're gay

[winter comet]

wonder what makes you think that 🤨

[astral surge]

fr

[astral surge]

Y'all pre calc is fun and all but this about it

[winter comet]

what?

[viscid thistle]

Pre-Calc is fun when you understand it ;-;

[rigid burrow]

Hi guys I'm stuck at getting the derivative of this function, can I get an idea how to proceed?

[willow skiff]

I wouldn't use the quotient rule to be honest

expand and divide

$y = 2x^{2/3} + 2 \sqrt{2} x^{1/2 - 1/3} + x^{-1/3}$ then power rule

[obsidian monolith]

higher's secret twin brother

[rigid burrow]

Thank you!

[willow skiff]

no problemo

[tame pine]

Yo

[whole flame]

Hello guys
I need references for which book I can use for Maths for class 12th boards?

[quasi elbow]

can someone please explain the concept of composite functions to me?

like, for instance, how is sin(x^2) a composite function?

[uncut mulch]

you have
trigfunction(something that isn't just x, )

[quasi elbow]

how does it make it "composite", though?

[uncut mulch]

function of (some other function of x)
its composite by definition

f(g(x))

[quasi elbow]

oh so like, x^2 is a function of x?

[uncut mulch]

yes

[quasi elbow]

got it! thank you 🌸

how are we getting this?

[quasi elbow]

according to this,

should not this be non-periodical?

because pi is irrational

[willow skiff]

no, this is periodic cause it's a rational multiple of pi

[quasi elbow]

then why is f(x)= |sin(pi x)| + cot(3x/2) not a periodic function? if its period is coming out to be 3/2pi

[willow skiff]

cause the period of |sin(pi x)| is 1
the period of cot(3x/2) is 2pi/3

the problem is that 1 is not any rational multiple of pi

or 2pi/3 is not a rational multiple of 1

[quasi elbow]

stupid question

what does one mean by rational multiple?

[willow skiff]

for example, -2pi or 3pi/4 or 1327pi/2093 and so on

are all rational multiples of pi

[willow skiff]

so I mean $\frac{p}{q} \pi$ where $p, q \in \mathbb Z$

[obsidian monolith]

higher's secret twin brother

[quasi elbow]

then why is 1 not a rational multiple of pi?

i am sorry i am just really bad at it

[willow skiff]

if 1 were a rational multiple of pi

then we could write $1 = \pi \frac{p}{q}$ or $\frac{q}{p} = \pi$ where $p, q \in \mathbb Z$

but pi is irrational

contradiction!

[obsidian monolith]

higher's secret twin brother

[quasi elbow]

right

so

[quasi elbow]

this is not necessarily true?

[willow skiff]

no, that's true

the periods of sin 3x and |tan(x/2)| are rational multiples of pi

in fact you can see the combined period must be 2pi

that's the least common multiple of 2pi/3 and 2pi

[willow skiff]

the least common multiple thing is not always true

[quasi elbow]

makes sense, thank you

[viscid thistle]

I started pre-calculus a couple of days ago and it's damn hard, and I can't understand very specific equations that I need help 😭

[willow skiff]

it's true like 90% of the time btw

[willow skiff]

!helpp

[tender quest]

To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.

[willow skiff]

you can ask in a help channel

[glass cipher]

yo

Anyone knows how to solve this??

pls

[willow skiff]

#help-7｜zen1thxyz message

I answered your question here

[quasi elbow]

is there any other way to do this question?

[willow skiff]

not really unfortunately

you can at most eliminate options a and b, cause the answer will be positive and those options give a negative answer

and then it happens that option d is always greater than 1, use calculus or something
but 2 cos(A/2) < 2 cos(120/2) = 1

I think it's more trouble to do it this way than knowing how to simplify the given expressions

but if you are smart you can eliminate options quickly

[quasi elbow]

what do i need to learn before solving such type of questions? i think it does not really fall under the "half angle formulae" category and they have started by doing the questions itself rather than providing the concept(s)

[willow skiff]

you need to learn the technique of writing 1 as sin^2 u + cos^2 u

here u = A/2 cause we use the double angle identity on sin A

and I mean why wouldn't you try u = A/2 if you want cos A/2

[quasi elbow]

i mean i get you but

when i first looked at the question, my first thought was not to start with the thing written on the top right

[willow skiff]

yeah I mean when you have no idea how to get to cos A/2

it's worth simplifying one of the options since they are all so similar

these are just test-taking techniques

cause JEE and everything

not to do with actual maths in my opinion

[quasi elbow]

so, hypothetically, what if we chose option d then?

like how are we supposed to know which option to start with?

[willow skiff]

well using the trick in the screenshot, you would always get $|\cos A/2 + \sin A/2|$ and $|\cos A/2 - \sin A/2|$ for all the options

[obsidian monolith]

higher's secret twin brother

[quasi elbow]

i see

[willow skiff]

it doesn't matter which option you pick to start with

they're all nearlythe same

[quasi elbow]

yeah but you gotta choose one option

and you do not have time to pick a wrong one in the exam

[willow skiff]

so you choose all of them at once

if you get what I mean

[quasi elbow]

i do not

but yes, sort of

thank you though

[willow skiff]

np!

[cyan cairn]

valsl

[warped shuttle]

how can i find how many real roots does this have ?
(x^2 - 2)^2 - 9x^2 + 32 = 0

[viscid thistle]

expand and simplify the brackets

you'll get x^4 + something * x^2

then u can just make a new variable y, and say that y=x^2, make a new equation in terms of y and it will be in the quadratic form and u can just use the discriminant

[viscid thistle]

or you could also get it into the form a(r^2 - 2)^2 + b (r^2-2)+c

then find the solutions of this quadratic

you will be left with r^2 -2 = 1st root , r^2-2 = 2nd root
not you have 2 other quadratic

solve :))

if b^2 - 4ac < 0 then the whole quadratic doesn't have any real roots and on the other case
b^2-4ac >=0 we have the roots as real

DANG!

i would have never got this idea when i was in high school

it is correct, right ?

nvm
plex idea was way simpler

[warped shuttle]

now what, desmos shows there are 4 roots

ok nvm i thing i got it

[winter comet]

t=x^2, t = 9 or t = 4, so what are the four values of x?

[warped shuttle]

yea i realized

thanks

[winter comet]

xD

[warped shuttle]

i forgot t was x^2 lmfao

is there a faster way to determine only the number of roots ? cause expanding and solving for t takes time and it gives the values which i don't need

[winter comet]

what are you trying to do?

oh

[warped shuttle]

it's a question

how many roots does ... have

[daring tapir]

As many as the highest power of the polynomial

[winter comet]

but thats the maximum amount

there could be repeated roots

or imaginary roots

if those still count as roots then i guess yeah

[daring tapir]

Uuuuuuh i mean you could kinda differentiate and set derivative = 0 at the point but youd need to know the root beforehand

[daring tapir]

Imaginary sure but repeated ones are eh

[winter comet]

that would help you find repeated roots? how?

oh

[daring tapir]

Mmm

[winter comet]

if it changes i guess

[daring tapir]

Not sure how itd work with triple repeated roots

So like (x-1)^3 p(x)

In an expanded form

[winter comet]

i mean

[daring tapir]

Ah yes

[winter comet]

it switches, andt hen it switches again

so it just keeps going

cuz the number of repeated roots is odd

[daring tapir]

Oh

Change that to (x-1)^4 then

Howll you predict that one?

[winter comet]

then it would switch an even number of times

so it would switch

like if it was going up, now it'll go down

at x=1

[daring tapir]

How would you distinguish between 2 and 4 roots tho

It switches similarly

[winter comet]

that is a problem

;l

[daring tapir]

,w graph (x-1)⁴

[obsidian monolith]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by enderdoesntmind

[winter comet]

yea it switches at x=1

[daring tapir]

,w graph (1-x)^2

[obsidian monolith]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by enderdoesntmind

[winter comet]

also switches at x=1

:(

[daring tapir]

Uh can you predict how "fat" the graph is using derivatives

Nvm

[winter comet]

maybe concavity?

[daring tapir]

At this point itd be faster for them to just find all the roots lol

[winter comet]

more concavity -> wider curve mebbe?

[winter comet]

fr lol

[daring tapir]

I guess so

[winter comet]

hey yeah just use curvature :D

such great idea

[daring tapir]

Lets graph the derivatives

[winter comet]

💀

[daring tapir]

,w graph 4(x-1)^3

[winter comet]

bro forgot the "graph" part

[obsidian monolith]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by enderdoesntmind

[daring tapir]

,w graph 2(x-1)

[obsidian monolith]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by enderdoesntmind

[daring tapir]

Argh

[winter comet]

concavity is increasing

wait a minute...

[daring tapir]

,w graph 6(x-1)^5

[obsidian monolith]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by enderdoesntmind

[daring tapir]

I cant tell the power 3 and 5 ones apart

So kinda useless ig

[daring tapir]

Fat better

[daring tapir]

😭 let a guy struggle

[winter comet]

nah bro forgot the "graph" part like 4 times in a row 😭

[daring tapir]

It was just twice ayo 😭

[winter comet]

ssuuuuuree '

[river drift]

curvature of a curve $y = f(x)$ is [ \kappa = \frac{\abs{f''(x)}}{(1 + [f'(x)]^2)^{3/2}} ] the radius of the osculating circle is the inverse of curvature [ R = \frac 1\kappa = \frac{(1 + [f'(x)]^2)^{3/2}}{\abs{f''(x)}} ] which is probably what you would think of as a measure of ``fatness''

[obsidian monolith]

cloud

[winter comet]

isn't concavity...sorta similar?

[river drift]

well concavity gives you (twice) the leading term of the 2nd order taylor polynomial (basically the "tangent parabola") whereas the osculating circle gives you the radius of the osculating circle (basically the "tangent circle"). so both of them, in some sense, are the next level up from the "slope of the tangent line"

[winter comet]

ohh yea that makes sense actually 😂

probably curvature is more accurate then

[river drift]

https://www.desmos.com/calculator/juytvmp45q

[simple scroll]

hey can someone help me understand factoring?

i get it, but the questions they ask me on khan academy

I don't understand

[dark acorn]

x can be -1 I believe

-1-1 isnt 0, therefore it can fit in the denom

i think

right

x can be 0 tho

[simple scroll]

Can u explain?

[dark acorn]

that was wrong, basically the denominator cant equal 0 at any step in your simplification, thats where the restrictions come from

basically, x^2-1=/0 and x-1=/0

find values for x that make those equations equal 0 and those are your restrictions, aoi messed up by saying x=/0, when it can

so the answer for that problem is C i believe

[simple scroll]

X equals divided by 0?

[dark acorn]

no x cannot equal 0

[simple scroll]

Oh sorry 😭

I read it like that that

[dark acorn]

≠ this thing

loll

all good

[simple scroll]

Yeah

[dark acorn]

x^2-1≠0, x-1≠0

[simple scroll]

Restrictions, right?

[dark acorn]

yea

[simple scroll]

Hold on can I send another problem and maybe I can understand?

I still don’t feel like I’m getting it

[dark acorn]

yes go ahead

[simple scroll]

Ty

[dark acorn]

ok

[simple scroll]

soo like.. please break this down for me

[dark acorn]

she didnt state her restrictions

the simplification is correct

or wait nvm

[simple scroll]

can you explain the restrictions?

[dark acorn]

hold on sorry lemme look again

ok yeah

im pretty sure she just factored the numerator wrong

(x+2)(x-2)= x^2-4

[simple scroll]

how so?

[dark acorn]

its a difference of two squares

[simple scroll]

oh it has to be two negative two's

[dark acorn]

uhmm

wdym

nvm yeah i think it does

it should be (x-2)^2

or (x-2)(x-2)

sorry its been a minute since ive done these😭

[simple scroll]

It’s okay

I don’t understand what they mean by excluding this number, not excluding that number, etc

[dark acorn]

excluding basically means stating what x cannot equal, so when she excludes x=0, shes saying one of the values of x cannot be 0

because if you put 0 in, the denominator would equal 0 and dividing by 0 is undefined

[simple scroll]

okay, but I probably have a test coming up soon and I can't assume what values do or don't equal to x

not saying that's your problem

but it's just making things more difficult for me to understand

[dark acorn]

ok lemme write this out rq

[simple scroll]

okay

[dark acorn]

its hard to explain over discord like this

[wraith void]

Denominator is equal to zero with x=2 not x=0

[dark acorn]

i know its just an example

[simple scroll]

when you go to multiply 4 and x, why leave 5 out?

[dark acorn]

wdym

[simple scroll]

also yeah subtracting 2 and 4 by themselves is gonna give them 0

what is that to prove?

[dark acorn]

restrictions only apply to the denominator

the numerator can be 0

[simple scroll]

oh sorry I was looking at the numerator and denominator at the same time

[dark acorn]

all good

[simple scroll]

so when the numerator is 0, does that mean the entire thing is just 0 at that point and the denominator doesn't matter?

[dark acorn]

basically

0/x is just 0

or 0/ any number

[simple scroll]

so undefined or still not equal to 0?

[dark acorn]

no undefined is seperate

thats why the denom cant be 0, when the numerator can

if im understanding your question right

[simple scroll]

yeah

so if it cannot equal to 0

wait okay

so if you cancel out, right?

its like canceling them out, if you were to divide them, it would be equal to 1

so is that another way of saying whatever you factored out cannot be 1 either?

[dark acorn]

hmm

idk if id put it that way

unless im misunderstanding

if the denominator is 1-x, what can x not be

[simple scroll]

ig 1?

[dark acorn]

yeah

thats how i think about restrictions

what value of x would make the denominator equal 0

[simple scroll]

so if the numerator is 0, then it's 0

if the denominator is 0, undefined?

[dark acorn]

yes

not the restriction

the whole fraction

[simple scroll]

this is so confusing

i feel like dying

[dark acorn]

😭

ok so, the restriction is the value of x that causes the fraction to become undefined, or what causes the denominator to equal 0

[broken heron]

What's a restriction

Oh wait this is precalc

I'm in alg 2

[dark acorn]

its an alg 2 topic as well

[dark acorn]

tan(83.9)=x/69 for how high it is i think

[warm relic]

Does anyone know what this is? I can't find a good video to explain it

[still laurel]

Anyone pls help T_T

[willow skiff]

sin, cos, tan in all 4 quadrants

so $\sec^2 x = \frac{1}{\cos^2 x}$ and so on

[obsidian monolith]

higher's secret twin brother

[willow skiff]

https://www.youtube.com/watch?v=V8LEHEzdU2U

[willow skiff]

you just need the equation of the line passing through (-5, 1) and (-3, -2)

and the line through (1, -4) and (3, 1)

[uneven nexus]

What's wrong with this way

I got infty by this way

[willow skiff]

you can only split the limit like that if both limits exist

here you have a $0 \times (-\infty)$ form, so that is undefined

[obsidian monolith]

higher's secret twin brother

[willow skiff]

for example, take $\frac{1}{n} \cdot n$ and $\frac{1}{2n} \cdot n$ as $n \to \infty$

[obsidian monolith]

higher's secret twin brother

[uneven nexus]

Ooh

[willow skiff]

there are multiple ways to approach the question, of course LH is possible

(infinity/infinity) works for LH too

or you could split the numerator as $\ln(e^{2n}) + \ln(1 + e^{-2n})$

and $\ln(1 + e^{-2n}) \approx e^{-2n}$ as $e^{-2n} \to 0$ by Maclaurin series

[obsidian monolith]

higher's secret twin brother

[uneven nexus]

Brilliant! Thx

[willow skiff]

nwnw!

[tame pine]

Yo

southy

[warped shuttle]

how to approach this question ?
in the sequence 1,4,7,10,... what's the minimum amount of elements we need to add that starts with the first element until the sum is bigger than 400 ?

so it means something like this
a1 + a1 + a2 + a1 + a2 + a3 ... > 400
which means
S1 + S2 + S3 + ... + Sn > 400

[willow skiff]

yo

[tame pine]

Yo bro

Imma come after some time

Ok doing an assignment

[quasi elbow]

what does this mean?

[agile sparrow]

(n) - Number of terms

[quasi elbow]

no, what's the bar over it?

[tame pine]

Where are my bros

[drifting crag]

@ionic locust

[drifting crag]

@ionic locust

[ionic locust]

.

[quasi elbow]

bump

[tawdry pilot]

a typo maybe

i’ve never seen something like that

maybe it’s trying to say beta / (n-1)

[agile sparrow]

Bracket (n-1)

[quasi elbow]

I just checked again from another website and it's simply (n-1).beta; I do not know why would they write it like that though

[quasi elbow]

got it!

[quasi elbow]

how did we get the second step of this?

[willow skiff]

product to sum formulae

oh wait they didn't use that they just observed that cos 60 = 1/2

[quasi elbow]

i got it, thank you!

[daring tapir]

Cos 20 * cos 40 * cos 80 = sin160/ 8sin20

But sin160 = sin20

So this is just 1/8

This * cos60

Gave 1/16

Just an alternate solution

[quasi elbow]

makes sense, thank you so much

[quasi elbow]

what formula are we using here again?

[daring tapir]

cos(x) cos(2x).... cos((2^n)x) = $$\frac{sin(2^{n+1}x)}{(2^{n+1}sinx)}$$

[obsidian monolith]

Ender Doesn't Mind
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[quasi elbow]

thank you!

[willow skiff]

very very handy identity

[daring tapir]

True

Its saved my ass more times than i can remember

[quasi elbow]

are we supposed to solve it like this or no? if yes, how do I proceed further?

[viscid thistle]

why do you just take 2 tanA to the left side giving you tanA=-tanB

[quasi elbow]

i do not know how that will help though

i tried this: