geometry-and-trigonometry

[craggy relic]

What you did with the modulus is wrog

[spring lion]

Oh wait

32

[craggy relic]

Yes

[spring lion]

Lol I just did the same thing with argument my bad

Ok

[craggy relic]

The e^ix thing is just another way to write cis x. I forgot people write it that way too

[spring lion]

Ok so...now I calculate?

Is that it?

[craggy relic]

Yes

Yep

[spring lion]

Google time bro

Let me see

[craggy relic]

You should be able to get the exact value of those but the problem made your life more annoying by asking for approximates lol

[spring lion]

Yessir I got it right

There is also this kind of questions

[spring lion]

😭 YOU TELL ME

I get something like

9.203

And the answer is?

9.2034

IMAGINE

(Just an example with the numbers)

[spring lion]

How do I deal with this?

[craggy relic]

Do you know what the roots of unity are

As an example it took me 0.5 seconds to rule out (a) from being correct. Why?

[spring lion]

Huhh

I'm afraid I have legit no idea

[craggy relic]

(a) is already in polar form

What happens when you cube it

[spring lion]

Polar cube

[craggy relic]

What's the resulting argument?

[spring lion]

Idk...

[craggy relic]

I mean the number inside the cis

Same trick as last problem

[spring lion]

Oh

Ohhhh

Well

3 times the og argument?

[craggy relic]

Yes

Which is

And where is that in the real complex plane?

[spring lion]

Wdym

[craggy relic]

So 3(60) is 180 degrees

[spring lion]

Yes yes

[craggy relic]

Visually where is 180 degrees on the real complex plane

[spring lion]

Oh

Literally on the real axis?

[craggy relic]

On which side

[spring lion]

Quadrant 2

[craggy relic]

Positive or negative

[spring lion]

Oh

[craggy relic]

It's not really in a quadrant since it's on the axis

[spring lion]

Ahhh

Mb

[spring lion]

Cos or sin?

[craggy relic]

The actual value is real because it's on the real axis

You should be able to tell whether it's positive or negative based on whether it's in the left half or the right half

[spring lion]

OHHH it's negative

[craggy relic]

Yes

And why is that bad?

(very simple answer)

[spring lion]

😭💔😭😭😭😭

Idk

[spring lion]

Why?

[craggy relic]

What do you want z^3 to be, as given by the problem

[spring lion]

Complex solutions

[craggy relic]

No

That's z

What is z^3

(it's given in the problem statement)

[spring lion]

3?

[craggy relic]

No

[spring lion]

Oh wait

27

My brain is fried man

[craggy relic]

Yes

Now the cube of whatever value a was is negative

But we want 27

[spring lion]

Yeah okay

[craggy relic]

So a is definitely wrong

[spring lion]

Ok

Now we try 120

[craggy relic]

More generally you need the argument of the solution, when multiplied by 3, to be a multiple of 360 (so that it lays on the x axis)

[spring lion]

360

[craggy relic]

Right so that one works

[spring lion]

Okay

[craggy relic]

Of course you need to check the modulus too

[spring lion]

They're all 3 here

[craggy relic]

But you should be very easily able to tell me what the modulus of every solution is

Yes

Ok so you literally only need to check the arguments

[spring lion]

Okay

[craggy relic]

There's two nonzero arguments that with

Work

[spring lion]

180 and 240 left in the answers let me think...

180*3

540

Nope

240*3

720

Is it 240?

<@&268886789983436800>

[craggy relic]

(under the equivalence of differing by a multiple of 360)

[spring lion]

Yeah

WOW the site just randomly refreshed itself and switched the question 😭

That sucks man

[craggy relic]

Gg

[spring lion]

Hahaahhaah

Well I know how to solve them now

[craggy relic]

240 was correct

Now comprehension check

[spring lion]

Okay

[craggy relic]

What are the 5 5th roots of 1?

It's exactly the same question, using 1 so we don't need to worry about the modulus

[spring lion]

💀

Let me think

1 times

Idkkkkk

[craggy relic]

You're looking for what angles times 5 are a multiple of 360

[spring lion]

Oh

72?

72 144 216 288 360(1)

[craggy relic]

And if you find the smallest nonzero one the rest will easily follow as multiples of that one

Like how 240 was just 2(120)

[spring lion]

Ahhh

I see

[spring lion]

Man I suck I can't solve this rn

[craggy relic]

So this is exactly the same thing, we want numbers that when multiplied by 4 are multiples of 360.

The easiest one is the one that when multiplied by 4 is exactly 360. Which is?

[spring lion]

90 180

[craggy relic]

Yes

There's one more

[spring lion]

270?

[craggy relic]

Yes

[spring lion]

Okay but the answers are just

-2i and stuff

[craggy relic]

There's a reason for that

What is cis 90?

[spring lion]

👁️

[craggy relic]

Just look at where it is on the complex real plane

[spring lion]

Um

Idk 💀

They actually didn't really use the word cis

If they did I somehow forgot

[craggy relic]

cis 90 means cos 90 + i sin 90

[arctic folio]

90 degrees is i no?

[arctic folio]

Ah

[craggy relic]

Yes

[spring lion]

Oh

Sin90 is 1

[craggy relic]

Yes

[arctic folio]

Cos90 is 0

[craggy relic]

But visually you can figure this out by thinking about where 90 degrees is on the real complex plane and then it's obviously just i

[arctic folio]

So cis 90 is just 1i

[craggy relic]

So that's why the answers don't have any real component

The real complement is just zero

[spring lion]

Ohhh so it's on the imaginary part of the complex plane?

[spring lion]

Right

[craggy relic]

It's on the imaginary access

Axis

[spring lion]

Yeah axis

[craggy relic]

My auto correct is popping off

[spring lion]

😭

Okay so

[arctic folio]

Uhhh

[spring lion]

It's 2 and 4 but negative or positive?

[arctic folio]

I still dont understand how any of this relates to the original z⁴=16 💔

[craggy relic]

The modulus is always going to be 2

[spring lion]

But where did we get the numbers 2 and 4 in the answers

[arctic folio]

Isn't i⁴ indefinite?

[craggy relic]

Can you show what you are looking at in the answers

[craggy relic]

What no

[arctic folio]

WHAT

[craggy relic]

Just square it twice

[craggy relic]

Well the answers that have a 4 are just wrong

The modulus has to be 2

Because 4th root of 16

[arctic folio]

Is there some other reason as to why my calculator gives i⁴ an automatic math error?

[craggy relic]

The cis 90 and cis 270 ONLY tells you the argument

[craggy relic]

Idk ur calculator sucks maybe

I have no idea why it would

[arctic folio]

It is an older model

[spring lion]

.

[arctic folio]

I'm just going to assume it sucks butt at non real equations

[spring lion]

👁️

[craggy relic]

It's a 4th root of 16

There for the modulus is the 4th root of 16

Aka 2

4 is bait

[spring lion]

Oh

You mean like

2⁴

Or wait

Nah I'm confused

[arctic folio]

Actually yes that is what they mean, only 2 or -2 to the power of four give you sixteen

[craggy relic]

The original question says z^4 is 16

[arctic folio]

Therefore anything with anything else is wrong

[spring lion]

Oh.

[craggy relic]

Hence the answer is a and b

[arctic folio]

Yep

[spring lion]

I see thanks

Can I send one last question before I just stop lol

[craggy relic]

Sure

[arctic folio]

Yesh

[spring lion]

I'm dying it's been almost 4 hours of studying now

[arctic folio]

Im almost on hour 2 but I'm having fun so far

[craggy relic]

This is similar to the first question

[spring lion]

It's incredibly fun yeah

[spring lion]

Yep

[craggy relic]

And once again it let's you avoid doing any real work because the hint tells you the polar form

Actually it's also similar to the second question lmao

[spring lion]

So the modulus

Is it 27?

[arctic folio]

Uhh

[craggy relic]

Yes

[arctic folio]

I haven't taken anything modulus related yet tbh

I have no idea what that is

[spring lion]

And the sin cos are 360?

[thorny venture]

is this precalc

[craggy relic]

Magnitude

[craggy relic]

Yes

[spring lion]

Yeah but trigonometry so I posted it here

[arctic folio]

???

[thorny venture]

nah ion care abt that part i was js curious

[spring lion]

Wtf is sin 360 🙏

[thorny venture]

cuz i recognized the question

i js finished precalc

[spring lion]

Niceee

[craggy relic]

Where is 360 degrees on the real complex plane

[thorny venture]

sin(0)

i think

i might be high

[craggy relic]

No you're correct

360 is is same as 0 for trig cuz you wrapped around the entire circle

[spring lion]

Isn't it same as 0 degrees?

[slim plinth]

I would just cube it

[thorny venture]

yeah

[arctic folio]

Yes

Lmao

[spring lion]

Okay

[craggy relic]

Too much work

[spring lion]

So

[craggy relic]

And not what the problem is trying to teach

[slim plinth]

Pascal triangles comes in play

[thorny venture]

i love this

this is so goated

[slim plinth]

(a+b)^3

[thorny venture]

yo what lesson is this

[arctic folio]

That's my problem, I'm good at mentally solving stuff but it usually takes me too long

[thorny venture]

i never did ts

[spring lion]

27*(cos(0)+i sin(0))?

[arctic folio]

Yeah that's not gonna be short

[slim plinth]

Let me try

[craggy relic]

Yes and what does that give

[spring lion]

Complex numbers

[arctic folio]

Not yet

[spring lion]

Let me calculate that

[thorny venture]

ywah but which part

ik ita complex numbers

[craggy relic]

It should be instant if you think about the real complex plane

[slim plinth]

(a+b)^3= a^3+3a^2b+3ab^2+b^3

Try that

[craggy relic]

0 degrees is on the real axis

[slim plinth]

Plug in numbers

[arctic folio]

Sin(0) is just zero so its just 27(cos(0))

[spring lion]

27?

[craggy relic]

Is

[spring lion]

Just plain 27?

[craggy relic]

Yes

[thorny venture]

yeah

[spring lion]

Okay

[craggy relic]

Hence why I said it's also like question 2

[arctic folio]

Yep

[craggy relic]

You already saw that exact number when you did question 2 lmao

[arctic folio]

So modulus 3 being magnitude means that if variable is squared then so is the modulus?

Am i getting that right

[craggy relic]

correct

normally i would write these as re^ix so (re^ix)^3 is very obviously r^3re^i3x

which is the same thing as the cis notation but makes the properties of exponentiating 10x more obvious

[thorny venture]

oh yeah

youll be flying through the next unit

[slim plinth]

Wait ima do the question you have us

By using pascal triangles

[craggy relic]

u really want to be leveraging your visualization of the real complex plane for this stuff cuz the unit circle is also a good way to visualize cos/sin as the x/y components

[spring lion]

Lmao hopefully man

[thorny venture]

nah i believe in you it’s easy

[arctic folio]

Would you recommend khan academy for someone trying to learn stuff on their own?

[thorny venture]

i’m going through pre calc again rn too

[spring lion]

Absolutely

[thorny venture]

yeah

[craggy relic]

it seems to be the #1 recommendation

[arctic folio]

I want to study stuff on my own during the summer

[thorny venture]

carried me so much

[craggy relic]

i would supplement it with aops if you wanted harder stuff

[arctic folio]

Hmm well i just wanted to make sure it's not like duolingo

[thorny venture]

what from apps

aops*

i’m tryna get better at comp math

[craggy relic]

aops has a few good textbooks that are good for getting into comp math

theres some other ones too, esp for getting into harder olympiad problems

[thorny venture]

i have the intro to geometry i think

cuz i suck at geometry

[craggy relic]

yeah their intro to geometry will teach u much more than an avg geo course

then to go further u can do some of their other books + similar ones

i suck at olympiad tho so i cant really rec too much specifics

[arctic folio]

Any.... Free recommendations?😅

[thorny venture]

alrifht i’ll look into it

yo you think you can help me with this problem rq none of the vids make sense

[thorny venture]

i think there is a few

on the website

not sure tho

[craggy relic]

try multiplying the vector [1 1] by that matrix

what do u get?

[thorny venture]

wait

i’ve only done multiplying mar mies by scalars

is it js the same thing or what

[spring lion]

Spoiler tag this

[slim plinth]

I get this as z^3=

[craggy relic]

i dont think u can understand what the matrix does if you do not know what multiplying matrices by a vector does.

[spring lion]

I didn't get there yettt

[craggy relic]

dont worry its a junk chapter

[thorny venture]

matrices make absolute no sense

to me

[spring lion]

Damn

[thorny venture]

ts the hardest part of precalc for me

[craggy relic]

ok

so what is a matrix

[slim plinth]

Hey Icarus what’s the ans choice

[craggy relic]

and wht does matrix multiplication mean

[thorny venture]

isn’t it used in lin alg

[craggy relic]

yes, but you learn it properly in lin alg

[spring lion]

To which question

[craggy relic]

thats why a junk short chapter on matrices is near useless

[thorny venture]

isn’t it just an arrangement of nunbers into like rows anr stuff

or like an organization of nunbers

[spring lion]

.

[slim plinth]

This one

[spring lion]

It's 27

[craggy relic]

but that wouldnt explain why multiplying matrices is such a convoluted and artificial process

[slim plinth]

Then I get this

[thorny venture]

yo no big boy words here please it’s a math server not english😭✌️

[spring lion]

Convoluted means like

[thorny venture]

i havent gotten to matrix multiplication

[spring lion]

Not simple or something

[thorny venture]

oh

[spring lion]

Like that

[thorny venture]

thanks

[craggy relic]

and you dont know how to multiply a matrix by a vector either?

[thorny venture]

nope

[spring lion]

Let me google it

[craggy relic]

then i dont understand why they are asking you this question

[thorny venture]

that’s all ik

for matrix anf multiplying

[spring lion]

Ok yeah it's hard complex etc

[thorny venture]

thanks

[spring lion]

[craggy relic]

i dont think you can understand the geometric effect of it if you dont know what it does

this particular matrix turns [a b] into [2a b]

[thorny venture]

what do you suggest i do

[craggy relic]

my personal suggestion is that touching matrices is completely unnecessary pre-linalg but that is probably controversial

anyhow you could just search what the rule is for matrix multiplication

[arctic folio]

Convoluted is NOT a big boy word omg 😭

[thorny venture]

yo chill on me

i suck at english

[arctic folio]

It's fine

[craggy relic]

the sentences people come up with complex math is much worse than tough english words

[thorny venture]

even though ts my main language

[arctic folio]

If you have time in the summer just reading a few books OUT LOUD and making sure you can sound out words is enough to greatly improve your abilities in english

[thorny venture]

reading is boring tho

[arctic folio]

It doesn't have to be an IRL book either, tons of free PDFs all around

[thorny venture]

i’ve been rryna read for like 3 years

[thorny venture]

yo

i lile found my own way of understanding kinda

[arctic folio]

Well if it's so boring you can't read consistently then that means you're probably addicted to your phone ngl

[thorny venture]

i am✌️

[craggy relic]

reading is boring

[thorny venture]

thank you

obly thint i kinda liked reading was shakespeare

[arctic folio]

Eh

I dont think it matters if you find it fun or not

[thorny venture]

that’s the only thing i’ve enjoyed reading in the past 5 years

[craggy relic]

clearly sitting and staring at the same math exercise for hours is superior

[thorny venture]

yes

it is

[arctic folio]

But it shouldn't be so boring you CAN'T do it

[rapid valley]

The last time I sat down and actually read something was an article for a 100 word essay on polar form

[arctic folio]

I honestly don't read too often either and I don't like most narrative based stories. My favorite book is one called "Lady Luck"

It's a basic book about probability

[thorny venture]

yo how would i do ts

[rapid valley]

Real mathematicians don't read

[craggy relic]

fk that

[thorny venture]

do i js skip it atp😭✌️

[craggy relic]

anyway i would just multiply the corners by the matrix

and see what it looks like

[dark sparrow]

does the word "determinant" sound familiar at all to you

(not discriminant)

[thorny venture]

yeah i forgot what ts is tho

i did matrices a long time ago

[craggy relic]

they teach determinants before matrix multiplication?

[rapid valley]

L quadratic formula reference

[dark sparrow]

https://www.youtube.com/watch?v=Ip3X9LOh2dk

[dark sparrow]

why L

[rapid valley]

That formula should be used in every example

[dark sparrow]

??

[rapid valley]

Nevermind

[thorny venture]

wwit

is it js

[rapid valley]

Jokes over

[thorny venture]

multiplying

the opposite sides

and subtracting

opposite corners or wtv

[craggy relic]

oh i didnt evenr ealize the question was asking for area

[dark sparrow]

you find the determinant of your matrix, and the area of your pre-transformation rectangle,

and then multiply those together.

[thorny venture]

so

the determinant in this case

is

-23 right

but area csnt be negative so i’m assuming absolute value of that?

[dark sparrow]

yes, you take the absolute value

yes, det(A) = -23, that much you're correct about

[thorny venture]

so js 23 x 40?

thanks

[dark sparrow]

so it seems, yes

[thorny venture]

what about this

would i js like multiply each thing by the determinant

wait

nvm

[craggy relic]

this is not asking for area

it is what i thought the previous question was asking

[dark sparrow]

the matrix

not the determinant

also consider watching the entire EoLA playlist (from which i linked ch6)

if you havent already

[thorny venture]

yeah i got it

i js made it into vectors

[mild jungle]

but i read

[distant umbra]

Multiplying a matrix by the vector is more or less like doing the dot product of the vector

For the area question you can assume one of the vertices to be the origin and write the other 2 vertices wrt that point

Multiply the two vertices by the matrix and you get the new vector

Area is cross product of 2 coinitial vectors

:)

[thorny venture]

whatd dot product😭😭😭

[dim trench]

hello im new here

[hollow lily]

Hi new here

[mild jungle]

lol

[distant umbra]

you havent learned dot and cross product before matrix transformations?

Geometrically the dot product represents projection

the basic logic is to multiply the coefficients of i, j, and k with each others cuz theyre perpendicular unit vectors

so dot of (1,2,3)(2,4,5) will be (2, 8, 15)

[thorny venture]

oh

i js do the multiplication

i don’t worry abt dot products and stuff

[distant umbra]

it is more of less the same thing

[gleaming mantle]

I have ideas on how to solve this equation, but in practice it turns out to be some kind of crap.

help

Okay, forget it, it's easier than I thought.

[hollow lily]

<@&268886789983436800>

[quaint chasm]

for triangulating any polygon, is the optimal result N-2 triangles?

[grave pond]

"Optimal"? If you don't add new vertices and divide the polygon completely into triangles, there will be N-2 of them.
(This can be seen either by considering angle sums, or by Euler characteristic).

[quaint chasm]

as in the minimum number of triangles
obviously there's several ways to do it

i.e you can segment any polygon into N triangles by just meeting them at the center

[grave pond]

That's why I said, if you don't add new vertices.

If you add M new vertices in the interior of the polygon, considering the Euler characteristic shows that there will be N + 2M - 2 triangles. That is minimized by having M=0.

[quaint chasm]

yeah i realized that as i said it

[lunar hull]

This only necessarily works for convex polygons.

Sorry, I am mistaken. It should be possible for concave polygons as well, it's just non-trivial to find such a triangulation

[quaint chasm]

yeah i was curious and tried it and it still worked

i just placed random lines until i got whatever side count

[trail tendon]

<@&268886789983436800> idk if you missed or are still on it lol

[exotic yarrow]

https://www.geogebra.org/calculator/f85x8rz6

[upper karma]

6 degrees of freedom on a sphere with 12 microthrusters

Atleast I hope.

All the T# are individual thrusters, 6 are on the -+xyz axes and the other 6 are +-45⁰ from the equator and the spread out in 120⁰ intervals. If anyone sees any problems please lmk😢

[cunning nebula]

so for translation and orientation?

[deep terrace]

This is trigonometry 😭

im in 8th grade trying to learn geometry, trig, alg 2, and precalc

prob best at geometr & alg 2

[timber kraken]

How are my answers?

technically this would be an equilateral triangle for number 6 because y=30 and angle C and angle B are equal so 60+2x=180 x=60

[deep terrace]

good. Excelt the last one is.. err…

[timber kraken]

what about it?

it’s the truth 🥺

[deep terrace]

im not sure what ur teacher would think...

kindergarden.. maybe like 3rd grade or 4th

[obsidian harness]

some kids actually learn this in kindy

somehow

[clear oak]

can I ask a simple qusetion here

math

[lost frigate]

sure

[lost frigate]

or is that a joke lol but if it is its fine

[timber kraken]

those kumon kids are getting too smart

[grave pond]

Suppose N>3, then choose a convex corner B with neighbors A and C.

1. If the diagonal from A to C is entirely in the interior of the polygon, then draw that.
2. Otherwise, let X be the point on BC that separates the part of BC that can be seen from A from the part that cannot. Then the line AX must pass through a vertex Y in the upper part of the polygon (if it passes through the middle of a side, there will be no points near X that A can see). So draw the diagonal AY.
   In each case, we have divided the original N-gon into two polygons with fewer than N sides; continue by induction.

Ah, no, that doesn't necessarily work, because AY might already be a side. Darn.

[lunar hull]

You can draw YB instead

[grave pond]

Ah yes, neat.

So either B can see some vertex in the dashed part of the polygon, or AC lies in the interior.

[upper echo]

By angle chasing ∠AMC=180°-x-(45°-x)=135°, so M lies on the circle centered at B of radius AB. Thus, ∠MBC=2x.

[junior jay]

Hey if I dm someone can you help me with geometry later on when school starts again?

My teacher doesn't teach in a way I understand

And school starts tomorrow

And we get lots of homework

[grave pond]

It's more likely that you'll get help if you just ask here.

People are generally reluctant to commit to helping in DMs.

[junior jay]

Alright ty sorry

[leaden tartan]

hello

[unkempt pebble]

Hi there, do you need any help?

[frail coral]

Hello could anyone help me understand my unit 5 geometry math lesson so I could pass my test with atleast a 80%

[unkempt pebble]

Hello

[frail coral]

Hello shammie

[unkempt pebble]

Yep ill try to help

Lay it on me

[frail coral]

Ok first would you want me to send the notes he gives us to study so you can see what the thing I need help with is

[unkempt pebble]

Sure then

[frail coral]

This is just the first one out of 5 he gives us

And we’re suppose to follow along in class but the way he teaches jsut really confuses me

That’s the second

[unkempt pebble]

I rarely teach math so brace for confusion

Alr then..

[frail coral]

Ok it’s all good

[unkempt pebble]

So..

[frail coral]

Want me to send all of them right now?

[unkempt pebble]

Hmm

Just dm me the rest or smth

[frail coral]

Ok

[unkempt pebble]

So basically you want me to teach geometry?

[frail coral]

Yea just this part

If that’s alright with you

[unkempt pebble]

Hmm...

Alr

[frail coral]

Btw I am retaking this test as I have got a 42% before and I have the test right now to do corrections with just to let you know

[unkempt pebble]

So... What chapter I might ask?

[frail coral]

This is the 5th chapter about inequalities in 1 and 2 triangles mediums and altitudes perpendicular angle bisectors and bisectors of a triangle

If that is what you mean

[unkempt pebble]

Lol I thought you needed help on a question

Not a revision teacher

[frail coral]

Oh lol

[unkempt pebble]

But I'll try my best

So, do you understand the term equidistant @frail coral

[frail coral]

Yes

[unkempt pebble]

Alr we're on to a good start

So, what part don't you understand?

[frail coral]

Ok the main part I don’t understand is figuring out how to do these types of problems like those types of ways

That’s what I didn’t know how to do most

But those units are 3 and 4

If I’m not mistaken

[unkempt pebble]

Oh alr

Some refer to the phytogras theorem

[unkempt pebble]

The best way in your case is to try and get a deeper understanding of the question

Maybe circle key points so you don't miss out

[frail coral]

Ok ok

[unkempt pebble]

And also, refer to the same chapter and revise that again and again and try to see what have you missed

[frail coral]

Ok I’ll try that out

[unkempt pebble]

I'm kinda surprised that you figured out the triangle pce contains the phytogras theorem lol

You're off to a good start for now

[deep terrace]

i learned that in first

in enrichment or something

we learned about squares, rectangles, triangles, and paralellograms

[dense matrix]

Where can i learn conic sections and coordinate geometry involving them? Like every derivation and proof included

[shut thorn]

A book i think.

[obsidian harness]

#book-recommendations

[valid shard]

Why is this not correct am I dumb

[obsidian harness]

also haven't you heard of SL Loney's coordinate geometry book?

[obsidian harness]

what's your u and dv there?

[valid shard]

Now I have it I make ln x my u and x my doc

Dv

Then it is correct I think

[obsidian harness]

can you show your work now?

[valid shard]

Yeah wait a moment

??

[valid shard]

Is it correct

[obsidian harness]

yep, correct

next time ask in one of the help channels, cause this is not the right channel

you're in #geometry-and-trigonometry not #calculus

[valid shard]

Okay I will do it

Thank you

[grave pond]

<@&268886789983436800> waver bot

[dense matrix]

i have now

[queen venture]

my notes

anyways parabola is so much fun than circles now that im learning it

[swift obsidian]

can someone help me? trig fucks my brain up, in this question you have to prove LHS = RHS and you can't assume it to already be true

[exotic yarrow]

Since you have two terms on the left but only one term on the right, I'd start by combining the two terms on the left into a single fraction.

Gtg, but if you need a further hint, consider ||how sec and cosec relate||.

(Spoiler alert: ||pythagorean identities||.)

[queen venture]

you dont have a choice

you got to memorize these

always multiply by the numerator on both numerator and denom

A normal to parabola y^2 = 4ax makes an angle 60 degrees with the line y = 4x - 3
Is it possible for it to pass through the focus?

[swift obsidian]

this tip saved me 😭

[exotic yarrow]

[deep terrace]

whats that.

[exotic yarrow]

https://en.wikipedia.org/wiki/Pythagorean_trigonometric_identity

[boreal shale]

what relation does angle 5 and 2 have

[exotic yarrow]

Corresponding angles

https://www.chilimath.com/lessons/geometry-lessons/corresponding-angles/

[boreal shale]

okay thank u man

[exotic yarrow]

[boreal shale]

can u pls explain how they are same side interior angles

bc i dont get how they are on the same side

[exotic yarrow]

[https://flexbooks.ck12.org/cbook/ck-12-basic-geometry-concepts/section/3.6/primary/lesson/same-side-interior-angles-bsc-geom/]

,rotate -90

[boreal shale]

oh so i gotta look at them from different perspectives

thx

[exotic yarrow]

[boreal shale]

yo would these be right

[cosmic igloo]

Yeah, angle 3 and 5 are corresponding if the lines are parallel. Angle 2 and 4 have no clear relationship without more info.

[craggy relic]

there are no two liens there that can be parallel they all intersect

[long geyser]

Are you allowed to construct lines in geometry like I did here with these 2 red lines?

This is the original square, for reference.

[exotic yarrow]

reposted here: #help-28 message

[boreal shale]

ok so they would be right?

[boreal shale]

ik but they use a different way to grade for some reason

like in another practice quiz i did

those would be counted as corresponding

im still rlly confused tho

[boreal shale]

oh btw corresponding angles can still happen even if angles arent congruent

ok ye i found out those are corresponding angles

[boreal shale]

would 2 and 4 technically be considered same side exterior angles?

[dense matrix]

would you be so kind to share them?

[queen venture]

you wouldnt understand them(i wrote them as per my need)

because all the derivations wont make sense as i just did them without saying what was the thought process behind them

[dense matrix]

alright

[upper karma]

Is it rhs = lhs

[orchid rampart]

Does it say prov this

[upper karma]

Hey guys

Can someone js lemme know how do I get a volume of a cone right circular, and based on diameter

Can just let me Kano if this is right?

Know*

NVM INGOTNOT

I GOT IT

All we gotta do is find d/2
then 1/3x22/7xd/2^2xh

[covert quest]

$$\cos ^{2}x+\sin ^{2}x$$ is this always 1 no matter what x is?

[somber coyote]

Totalani

[orchid rampart]

X should be real

[deep terrace]

they are corresponding. if line m was paralell to line k they would have congruent measures.
in this image you can prove m5 is not congruent to m2 because lines k and m intersect meaning they cannot be congruent, therefore proof by contradiction

yes

then also explain what lhs and rhs is

[deep terrace]

wtf

[shut thorn]

yes it doesnt matter what the angle is.

[deep terrace]

i CANNOT read that.

also waht does lhs and rhs mean

left smthn right smthn?!

[deep terrace]

i know

D:

my computer is really glitchy

not my eyesight

[cosmic igloo]

yep

[serene nova]

gulps,,, it would be so sigma of you guys.. if uou double check..🥹🥹🥹 went too far with A level I genuinely forgot how to do simpler questions my brain feels like its frying 💔💔💔💔 screeches in agony I FEEL LIKE A FETUS AAARRRGGHHGHHG

[strange pond]

you can construct pretty much anything on a diagram as long as you think it'll help you. for your information, such lines are referred to as auxiliary lines.

[long geyser]

I do appreciate the help. I Googled it after I asked this, and I found the same result.

[strange pond]

looks good

[serene nova]

OKAY PEAK

Got itttt

[jaunty jasper]

I’m confused for both these parts. Specifically getting the number ratio in part A and then just part B as a whole

[exotic yarrow]

ratio just means to divide

so for the corresponding sides, you'd divide the lengths of both sides in the pair

for the ratio of the perimeters, you'd divide the perimeters

for part b, pick some values for a,b,c

it'll be easier to see what's going on that way

[dark sparrow]

can you write down an expression representing the perimeter of FGH?

[upbeat violet]

I don't know how to reach the result. Can anyone?

[queen venture]

sure

[queen venture]

let the three points be

$$ at^2, 2at $$

[somber coyote]

fysch

[queen venture]

where t can have three values t1, t2, t3

now the equation of a normal is

$$ y + xt = 2at + at^3 $$

[somber coyote]

fysch

[queen venture]

if it passes through h,k then

$$ k + ht = 2at + at^3 $$

[somber coyote]

fysch

[queen venture]

ill write it as

$$ at^3 + (2a-h)t - k = 0 $$

now SA where A = (h,k)

$$ SA^2 = (h-a)^2 + k^2 $$

[somber coyote]

fysch

[queen venture]

SP

$$ P = (at1^2, 2at1) $$

[somber coyote]

fysch

[queen venture]

now distance of P from focus is as the distance of P from directrix

$$ SP = a(t_1)^2 + a $$

[somber coyote]

fysch

[queen venture]

the rest is left as an exercise for the reader

[somber coyote]

fysch

[queen venture]

this has three roots

$$ t_1, t_2, t_3 $$

[somber coyote]

fysch

[queen venture]

all the relations you will need will use vietta's theorem

$$ SP = a(t_1)^2 + a $$

[somber coyote]

fysch

[upbeat violet]

But how do we get RHS?

[queen venture]

look carefully

$$ a( (h-a)^2 + k^2) $$

[somber coyote]

fysch

[queen venture]

$$ t_1t_2t_3 = k/a $$

[somber coyote]

fysch

[queen venture]

$$ t_1 + t_2 + t_3 = 0 $$

[somber coyote]

fysch

[queen venture]

$$ t_1t_2 + t_2t_3 + t_3t_1 = (2a-h)/a $$

find h,k

[somber coyote]

fysch

[queen venture]

where was this prob from?

loney?

[upbeat violet]

Yeah

Thanks for the sol

[queen venture]

https://tenor.com/view/yagi-toshinori-all-might-my-gif-23944706

[jaunty jasper]

i have this, FGH being on top and XYZ on the bottom

[dark sparrow]

really overcomplicating

also not how you add fractions anyway

perimeter of FGH is a+b+c
perimeter of XYZ is 3(a+b+c)
their ratio is 1:3, that's it

[jaunty jasper]

oh thank you i get it now! My teacher just wrote it that way because it was something about ratios

[cobalt scarab]

I'm exited yall.
I've got my 1st taste of trig coming this semester!

[queen venture]

its boring

at the end

in start the rotating line is really fun

[shut thorn]

from our first trig chapter to our last one. The unit circle wasnt used once.

😭

for this grade ofc.

(10th)

there was a head bashing questions on the pythagorean identity

[rustic mauve]

10th 👍

[queen venture]

i was so confused in 10th

like how can an angle be 90 in a right angled triangle

and how can it be 0

the unit circle made sense afterwards

[shut thorn]

isnt it literally in the name? right angled 😭

[queen venture]

broski

sin90

cos90

[shut thorn]

ah

[queen venture]

in a right angled triangle

impossible to get other two angles to 90 or zero

[shut thorn]

eyah

i think that might be the reason this book avoids every question which involves sin90, cos90 in practical applications

[queen venture]

and they lowk the most ones

[shut thorn]

ok now im curious wait lemme see how they derived the value of sin90 and cos90 in our book

brb

[queen venture]

to use in physics

[queen venture]

u dont derive it

🥀

rotate the radius of the circle by 90

[shut thorn]

ye ik that way

[queen venture]

thats THE way

[shut thorn]

i think i dont know what the meaning of a derivation is

[queen venture]

thats how trigonometry is defined

in terms of coordinate geometry and this rotating radius

[shut thorn]

ye

alright so they didnt give how the values were found

they just said "fuck off. Here are the values"

[queen venture]

lol

[shut thorn]

axioms 😭

they are NOT axioms as much as i can think

[queen venture]

nahh

[shut thorn]

tbh i htink its just for a simplificatino

they called it axioms.

[queen venture]

yeah cuz they cant explain it

tbh trigo made no sense when i learnt it in school

i was confused what the teacher was cooking up

[shut thorn]

😭

now im scared for the next year when i inevitably learn trig.

in school

[queen venture]

u havent done it yet?

like the basic part

[shut thorn]

no ive taught myself using my sister's book.

[queen venture]

oh

i tried that in starting of 10th

[shut thorn]

approximately 14 crashouts on trig only.

[queen venture]

lwk got so bored

[shut thorn]

fire.

[queen venture]

now it feels algebra

🔥

[shut thorn]

i havent even done one full chapter of the geo section and i wanna pull my hair out because i cant even ask for help from teachers

theyre literallty just goanna say "why are you doing next year's syllabus" and send me the fuck off.

[queen venture]

self study is hard bro , its not very dopaminergic

cuz of the slowness

[shut thorn]

6 questinos in 2 hours

🔥

[queen venture]

i js get bored thinking when will the good part come

[shut thorn]

also we should move to chill before mods come.

[queen venture]

k

[misty cape]

u guys have any cool formula sheet for trigonometric equations

JEE

[shut thorn]

wasnt it pinned in hre

where did it go wha

[misty cape]

nooo not the pinned ones

i mean the other identities we use in trigno equations

i forogt them all

[shut thorn]

pythagorean identities?

here are the ones i know of. Wiat

[misty cape]

like sinx*sin(pi/3 -x)*sin(pi/3 +x)= something i forgot

like

usually in jee we remember this sort of identity cus its faster

[shut thorn]

alright i cannot

what 😭

[misty cape]

ur in 12th?

right?

[shut thorn]

no

thought i could help

[misty cape]

11th?

[shut thorn]

guess not

9th.

[misty cape]

uuuuuhhhh

ur freeking out cus i put pi

in a trigno

function

my bad gang

dont bother

[shut thorn]

no i know what it is

pi radians.

[misty cape]

yeee

ssoooo like we have a chapter in jee called trignometric equations

we learn some other miscellanous identities

and i kinda forgot them

u can search a oneshot for it and learn it urself prett fun

[shut thorn]

sure lemme find another way to pull my hair in frustration at 1 in the morning

would be fun

:D

[misty cape]

thing is its deleted in jee mains but it appears in questions sooo kinda need it

[orchid rampart]

Sleep

[misty cape]

math in jee is

funnnnnn

u'll have loads of hairfall

[shut thorn]

laterr

[misty cape]

ur in 9th

its bedtime

dawg

[orchid rampart]

Fix sleep

[misty cape]

ur 14

sleep

ima go sleep

gn

[shut thorn]

gn

[upper karma]

imagine coord bashing it

i accidentally used x^2=4ay instead of your parabola but its the same thing. In this solution i renamed A to T.

||let V be the vertex of the parabola, so that SV = a. Let the tangents to the parabola at P, Q, R form a triangle ABC like in the diagram.

Lemma: the foot from the V to any tangent to the parabola lies in the x axis.
Proof. In this proof we'll use the tangent from R. Let J be the foot of R onto the directrix. Then SR=RJ. From the optical properties of parabolas it follows that the tangent RA bisects <SRJ, so its perpendicular to SJ since SRJ is isosceles, but in particular it meets SJ at its midpoint, which is on the x axis because dist(V, x axis) = dist(direcrix, x axis). This is enough to prove the claim.

By the lemma, the feet H,I,G of S onto the sides of ABC are collinear, so S lies on (ABC).

Notice that the reflection of R over G lies on the y axis, and so do the reflections of P over H and the one of Q over I. Call these reflections D,E,F.
Let K be the reflection of T over S. Then from the previous observation we get that KD//TP, so D is the foot from K to BC. Similarly E,F are the feet from K to CA, AB. The feet from K to the sides of ABC are again collinear, thus K is on (ABC). Furthermore, since the simson line of K is the y axis and the simson line of S is the x axis, S and K are antipodes in (ABC) because their simson lines are perpendicular.

All this allows us to reduce the problem to showing that SD*SE*SF = SV*SK^2.||

||Now let d=SK.
(1) Observe that SV*SD = SH^2, so taking the cyclic product we get SD*SE*SF = (SH*SI*SG)^2/SV^3.
(2) A is the S-antipode in (SIG) and V is the foot from S to IG, so SV*SA = SI*SG. Taking the cyclic product we get SV^3 = (SH*SI*SG)^2/(SA*SB*SC).
(3) The reasoning in (2) applied to triangle SBC yields SH = SB*SC/d. Multiplying cyclically SH^3 = (SA*SB*SC*)^2 /d^3.

With these results we can finish.
Using (1), its enough to prove that SH*SI*SG = d*SV^2.
Taking everything to the third power and using (2) to substitute SV^3, after simplifying we find that we want to show that (SA*SB*SC)^2 = d^3*(SH*SI*SG), which is precisely (3) so we're done.||

Sex with parabola 🥵

[past path]

can i ask a question about arcs

Why is the answer to number 8 360?

[cosmic igloo]

Because arc CFB goes all the way around the circle, it’s the entire 360 degrees.

[past path]

yes but like arc CB is 20° so I figured that I would have to subtract 360 and 20

[deft bluff]

Yeah, whoever marked this is incorrect. Your logic is solid 👍

In order for the arc to go around the whole circle, it needs to start and end at the same point.

Indeed, in arc CFB, C is definitely not B :)

[deep terrace]

guys put this in desmos

• this for the cut off equation

Hint: It may or may not crash desmos for 10 minutes

[marsh widget]

Can someone tell me what answers they get for this cuz mine are always rounded wrng

Make sure u don’t round till the very end of the calculations btw

[dark sparrow]

can you share your answers first

[marsh widget]

Nvm I got the answer sorry

Thanks anyway

[sly urchin]

I proved that ABFC is harmonic

Now if I can prove A,X,F are collinear, i'm done

Someone help please

[upper karma]

proj spam even more

it's helpful to notice that ||PQ goes through T|| and ||BQ int CP lies on AF||

then you can do it in a lot of ways

[upper karma]

@upbeat violet I know you don't care but there was a part missing at the end, now it's fixed

[queen venture]

dont use this shitty sol

@upbeat violet

[upper karma]

Yours is shittier

[queen venture]

https://tenor.com/view/mahito-sukuna-itadori-laugh-risadas-gif-19649080

[upper karma]

Mine is synth you just bashed

[queen venture]

dont tell me you forgot vietta's

[upper karma]

?

You bashed

[queen venture]

nah

[upper karma]

Yes

[queen venture]

avg derive everything enjoyer vs js look it up enjoyer

[upper karma]

avg geo hater

I just think bashing is pointless

Why would you even do geo if you bash

but I know bashing was intended and your sol is not even that bad, I just posted mine for the funnies, and because it's more interesting

[orchid rampart]

fetish

[upper karma]

geo is life

[upper karma]

I found another sol inverting at S do you want it

It might be a bit shorter

inverting parabola with the ||2019 igo adv p5|| trick

[queen venture]

nah im not smart enuf to understand as you can guess

[queen venture]

its something i would want to get taught but id hate using it

[upper karma]

im just trolling

[upper karma]

why would you hate using inversion

inversion is the best thing ever

wait I misread

ok I'm just high

geo is worse than cocaine

[steady grove]

Can anyone pls solve question 60...

[sly urchin]

I did notice that P,Q,T was collinear and tried proving that using cross ratios but couldn't

I'll try to do something with your other hint rn though

Thanks a lot

[worn kite]

square root of 11/12

[deep terrace]

english

and maybe i can

[fallow wyvern]

how do u do this its trig

[lime crown]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[sly urchin]

I'm trying to use Pascal's theorem to prove the above observations, but i'm struggling
(Brocard's theorem for BCPQ is also interesting)

PS: I think i did it

[shut thorn]

since the triangles are right angled. Try to apply the definition of sinx,cosx, and tanx to the angles. For example. lets take x. Then tanx = opposite/adjacent.
Which is 12/14. Simplifying 12/14. We get 6/7 = tanx (now i dont know how to find the angle from here because i havent been taught that but you can put it in desmos?)
similarly try to do it for the other angles.

ok if i put tanx = 6/7 into a calculator then it gives an angle of 40.60 (rounded)

similarly. You can do the same with z and then use the angle sum property of a triangle to find w and y.

[past path]

Thank you 👍🏼

[past hearth]

if u know law of sines you could use it

[shell basin]

this is a statics problem but that isnt my issue my issue is that i think thatB is at (-2, 1, 2) but im not sure

[exotic yarrow]

I’d agree with you

[livid patrol]

Can someone help me please

[shell basin]

i did the grave sin of asking an AI and it kept telling me its (-1, 2, 2)

was too ashamed to admit it here but now that some time has passed yeah

[jaunty jasper]

Can someone help me? I got stuck

[exotic yarrow]

By the definition of similarity, we have
$$\frac{AC}{DC}=\frac{AB}{DE}.$$
What happens if you multiply both sides by $DE \cdot DC$?

[somber coyote]

Civil Service Pigeon

[jaunty jasper]

Once you multiply that gets you AC x DE / AB x DC by multiplication prop
and then switch it and that’s commutative prop?

[exotic yarrow]

If you want to be pedantic, then yes it's a mix of commutative and symmetric.

[jaunty jasper]

Okay thank you!

[exotic yarrow]

[spiral lodge]

<@&268886789983436800>

[sly urchin]

Do mixtilinear incircles have a nice representation in complex geometry?

[mighty monolith]

anyone have tips for geometry

my first semester grade wasnt as high as i wanted

i mostly need help for tests and how to study

[slim plinth]

Use 30-60-90 special right triangle

[shadow depot]

can someone help me with b

[shut thorn]

did it translate the "E" to "AND"?

[fossil glacier]

Oh shoot we got a geo champion from India

[lost frigate]

AH perpendicular to BC, how do i prove A,O and H align in a line? (dont use vector cuz it will be extremely easy, pretend i dont know about vectors)

andd uhhh not finding x, i found it anyways lol

[shut thorn]

You can do it using similar triangles if im sure

Draw HQ perpendicular to AC. Then from there two triangles are similar.

[lost frigate]

i think it only happens when A, O and H align, but we are proving it

[shut thorn]

Oh ye

Oh wait

Draw HQ perpendicular to AC. Then using the angle sum property of the quad OHQP where P is the perpendicular distance from the center of the circle. You get anglePOH = 180- angleAXQ
Since HQ is perpendicular to AC and so is OP. OP||HQ
-> angle AOP = angleAXQ
Now angle AOP + anglePOH = angleAXQ + 180 - angle AXQ = 180
Hence AOH is a straight line.

This should be correct?

i just realised that i called AHQ AXQ the entire time because the H is so much less visible than the X.

https://tenor.com/view/cry-blacknoob-waah-wah-ueee-gif-3513992683220693199

[sly urchin]

If O is the circumcenter of ∆ABC, what is the circumcevian triangle of O wrt ∆ABC called?

[sturdy leaf]

@north ginkgo one day rubiks cubes will allow us to time travel

[lost frigate]

X isnt a point (its a value we need to find but im not asking that tho)
but i see your point!
we will use trig there
imma say OP perpendicular to AB at P
by that BP=20-6=14
we look at triangle POB, we got tanPOB=7/3=>POB=arctan(7/3)
now we can calculate AOH=360-(90+2arctan(7/3))+45 approximately 178,6 degrees
they dont align lol, i just messed up with my eyes because the degrees are too close

[shut thorn]

Ok do i made s mistake sonewuere

[lost frigate]

take a look at this triangle, when i change size AB and AC, they dont align

[shut thorn]

Ye

[lost frigate]

so if you dont use AB and AC to prove, it means they dont relate, which is false

[shut thorn]

Did i fuck up a corresponding angles.

😭

[lost frigate]

u still did well tho, im bad at proving these stuffs ngl

[shut thorn]

Oh ny fuckn god

I took AOH as a transversal

AAAAAAAAAAAAAAAAAAAAA

Im goanna bite off my fingeri iwnwrjw

[lost frigate]

nah when u find mistakes u will able to find chance to be better

just dont do it in exams 🔥

[timber kraken]

How does my proof look

I also did this page

[shut thorn]

why is the light so blue

[shut thorn]

correct/ Absolutely correct

[silent plank]

question 2 and 5 are wrong

[silent plank]

^

[shut thorn]

in this one 2nd is wrong. thats a critera of RHS

one way you can prove that its a criteria of RHS is by using pythagoras theorem

[timber kraken]

huh

SSA and ASS aren't verification methods

therefore I wrote none

ion get it

whats wrong with 5

nvm i get 5

AAA isnt a verification and for 2 I should of used hypotenuse leg theorem

thanks guys

[timber kraken]

oh those are my leds haha

[shut thorn]

Why do yoy have leds. And blue ones of all colors

😭

[raw marten]

leds are peak and so is blue , don’t discriminate against blue leds😕💔

[placid estuary]

👋

[timber kraken]

lol

[shut thorn]

i dont discriminate against leds. i discriminate against lights in general

[timber kraken]

gasp

hence dark mode you have on rn?

[raw marten]

how are you gonna see without light 😨

Discriminating is for bullies 😕💔

[timber kraken]

solving long and scary proofs are for bullies 🐱

tsk tsk

[sacred canopy]

Hi, do you know about some free software where i can graph hyperplanes?

[timber kraken]

im struggling with geometry rn I barely know the characteristics of a plane but you can always open a fourm!!!!!!