help-0

[karmic rapids]

confused those

[marsh rapids]

I think I've seen a similar exercise not too long ago. Gimme a sec and I should be able to find a proof for this

[karmic rapids]

Prove the following (simplified) variant of Brouwer's fixed point theorem:
Every $f \in C(I, \mathbb{R})$ with $I := [-1, 1] \subset \mathbb{R}$ and $f(I) \subseteq I$ has a fixed point

[ocean seal]

illuminator3

[marsh rapids]

IVT should work for this quite easily

[karmic rapids]

my first question or the whole question

[marsh rapids]

this

[karmic rapids]

if IVT proves that then this must also be true

because lipschitz constant implication is both ways

[marsh rapids]

no

[karmic rapids]

huh

[marsh rapids]

or give a proper proof because I don't see what you mean by this

[karmic rapids]

nvm

[marsh rapids]

Here's a "tweaked square root" counter example

where f' also is continuous on I

[karmic rapids]

mh

alr

will try to do it with IVT

thx

.close

[lone heart]

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[iron forge]

hi

[alpine sable]

Start by multiplying E2 and E1

[iron forge]

sending my work one sec

did i multiply it incorrectly?

[alpine sable]

Yes

Now find its inverse as AB = I is possible iff A and B are inverse of each other.

You know about inverse?

[iron forge]

Yeah sorta

is my first step right though, the E2 x E1

[alpine sable]

Yes your multiplication is correct

,w inverse of [1 , 0][0, 1/7]

[ocean seal]

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[alpine sable]

You did E1 × E2 it has asked for E2 × E1

Ooo

Yes

[alpine sable]

Bruh there's no need of inverse it's unnecessary mess

[iron forge]

oh thats why

[alpine sable]

I checked multiplication but didn't check that.

[iron forge]

Alright im going to redo it thank you

[zenith carbon]

What’s the difference

You multiply the two numbers either way

[alpine sable]

They are not numbers

[alpine sable]

It's different in Matrices these aren't you friendly neighborhood numbers

[zenith carbon]

Ah

Ima just slide out of here

[lone heart]

@iron forge Has your question been resolved?

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[rocky dagger]

what would be an efficient way to approach this question

[rocky dagger]

this is what i have so far

im wondering what the fastest way would be

[remote heron]

wonder if you can bound them=

[rocky dagger]

wdym

[remote heron]

idk ill have to play with it

like if you can bound some cells say the two right above the central cell

leaving only a single option

[rocky dagger]

my friend did it by using constants for each box in the bottom row and working his way up

and then subsituting a bunch of sim equations

but thats long

[rocky dagger]

not sure what you mean 🤔

[remote heron]

it must be something about the double counting

right

like you can get the sum of the second row very quickly

because you know 16 gets counted twice

youll have a similar outlier or whatever you wanna call it with the central element in the last row

[rocky dagger]

wdym by outlier?

[remote heron]

like multiple times counted

they give you the sum of the rows

im just trying to track

i think this is the way

like

the difference between rows is the uhh

sum of the centers

right

so then the sum of the middle three elements in the bottom must bee

12

[rocky dagger]

hmm

[remote heron]

yea

i ithkn i got it

you wanna vc?

or i can type

[rocky dagger]

i cant talk

but yeah

[remote heron]

oh, so vc?

[rocky dagger]

yeah will type tho

[remote heron]

okay

[rocky dagger]

one sec

let me fix the screen

discord bugging in browser lol

ok got it

i got all of them besides 2nd last

maybe i couldve of if i continued but i felt it was pointless effort lol

yeah i got 45

yh

yes

yeah

ok

ok

ok yeah

oh i see

yes this seems really fast

so the main part was seeing the pattern that the mid is the difference

yeah this does seem pretty fast

thanks a lot

[remote heron]

[rocky dagger]

gonna try more problem solving questions!

might be back lol

[remote heron]

[rocky dagger]

!solved

hmm

[remote heron]

.close

[rocky dagger]

.close

[lone heart]

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[obtuse grail]

pls help me

[raw latch]

which step is troubling you?

[obtuse grail]

well

I did the problem

but im finding different answers online

3 websites w 3 different answers

so im second guessing myself

[obtuse grail]

step 3 mostly

[raw latch]

alright let me graph it and post a picture to talk about

[obtuse grail]

sounds good

thank you

[obtuse grail]

so I did step 1 I feel good about that

-4/5, 4/5, -1, 1

I graphed step 2

so im good on that

its step 3 and onwards where I need help

[raw latch]

alright

so what im thinking is they want the area of the region for negative y

[obtuse grail]

ok this is a fubini problem just so you know

so its like the type 1 type 2 thing

[raw latch]

i know fubini but i dont know what you mean

[obtuse grail]

theres two types of fubini

this is a type 2 by the look of it right?

[raw latch]

yes

[obtuse grail]

ok are there

ok this is the rubric for it

what is this

are there disjoint regions?

@raw latch pls help

[raw latch]

i mean these functions never meet

so there should only be one region

[obtuse grail]

okay got it

[raw latch]

the lower one

[obtuse grail]

what do you mean the lower one

[raw latch]

negative y

[obtuse grail]

is this middle part disjoint

[raw latch]

the part of the area for positive y is not bounded by anything

only for negative y

[obtuse grail]

thats right

i see that your right

now for step 4

what are the upper and lower limits

[raw latch]

what do you think

[obtuse grail]

@raw latch

[hazy island]

I’m new here

[obtuse grail]

@raw latch pls help

<@&286206848099549185> hello? please help me im stuck

<@&286206848099549185>

[lone heart]

@obtuse grail Has your question been resolved?

[lone heart]

@obtuse grail Has your question been resolved?

[lone heart]

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[lone heart]

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[bronze mauve]

If $|G:H|=n,$ find a homomorphism of $G$ into $S_n$ whose kernel is contained in $H.$

[ocean seal]

monkeman

monkeman

[bronze mauve]

oh wait

i'm dumb

so theres the obvious homomorphism

[ocean seal]

monkeman

monkeman

[minor nimbus]

Can anyone help me with some hw problems?

[worn fox]

#❓how-to-get-help

[lone heart]

@bronze mauve Has your question been resolved?

[bronze mauve]

ok so i figured this part out

this is the full problem that i am stuck on

i figured out the hint

[ocean seal]

monkeman

[bronze mauve]

<@&286206848099549185> i need help on a few things

i don't know exactly how to define an explicit homomorphism from $G$ to $\text{Sym}(K)$ and i don't know how this exactly connects to proving the original problem

[ocean seal]

monkeman

[bronze mauve]

ok never mind i'm trolling

.close

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[bronze mauve]

i'm just bad at math

:(

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[still storm]

hi, how do I show there's no global extremes for this fn?

[hazy plank]

what have you tried so far?

[still storm]

I have my critical points

[slate jolt]

which are?

[still storm]

(0,0,-1)

(1,-1,-1)

which (0,0,-1) is a saddle

and (1,-1,-1) is a local min

also I got there's no singular points as well?

@slate jolt hello?

[slate jolt]

oh sorry

isnt a singular point just related to curves?

what would be the definition of this (i didnt do differential geometry in english soi i'm not too familiar with the terms)

[still storm]

my definition says that a singular point is where the f(c) is not differentiable

at point c

[slate jolt]

oh ok

so yeah

you have now your only possible global extrem which is (1,-1,-1)

you must now find a point that is smaller than it

[still storm]

for global min?

[slate jolt]

yeah, its a local min

so for it to be a global min

it must be smaller than every value

but there are points at which f is smaller

just give one of them

[still storm]

ok

tx!

I got it

[slate jolt]

so the reasoning is:

if there is a global extreme it is also a critical point

you find all of them

examine the geometry nearby them

if there are local extremes

show that they're only local

[still storm]

cool, thank you for this, I know how to say now

tx

.close

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[alpine sable]

this might be a stupid question

[alpine sable]

but

a sequence has to be bounded in order to converge. so, if we want to prove that a sequence, say a(n) converges, we have to find a bound for a(n), so thats why we use cauchy's condensation test right?

[hazy plank]

I think it has to be more then just bounded

[slate jolt]

converging --> bounded

but not the other way around

sin(n) is bounded

cauchy's condensation is much stronger than saying that the sequence is bounded

[lone heart]

@alpine sable Has your question been resolved?

[alpine sable]

right right

[lone heart]

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[narrow junco]

can anyone help?

[lone heart]

@narrow junco Has your question been resolved?

[narrow junco]

hwo can u pls explain?

[wary stream]

Just ignore that user, they need to read #❓how-to-get-help first because they started asking a question in other channels

[narrow junco]

okok

[lone heart]

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[alpine sable]

How can I calculate an x,y point knowing only the radius and angle of a vector?

[wary stream]

Using sin and cos

[alpine sable]

those are basically black boxes to me

[wary stream]

To break it into x and y components

[alpine sable]

don't understand how they give us the x y components

[wary stream]

You find those

Using sine and cosine

[alpine sable]

sine is a ratio of r/x

[wary stream]

No

[alpine sable]

sorry r/y

?

[wary stream]

Yes to this

[wary stream]

No to this

[alpine sable]

how can we do r/y if we don't know what y is

[wary stream]

Actually, it's still a no

The ratio is slightly wrong

$sin \theta = \frac{y}{r}$

[ocean seal]

dldh06

[wary stream]

You have the angle and r

Find y

Same process for x

[alpine sable]

How do I find y?

[wary stream]

Using math

You have the angle, and r, like I stated

Use those to find y

[alpine sable]

I dont understand how only knowing the angle and r can give us y, unless we just manually pinpoint it on the graph with our eye balls

[wary stream]

Do you know how to solve something like 3 = x/65?

To find x

[alpine sable]

multiply both sides by 65?

[wary stream]

Yes

[orchid bison]

yep

[wary stream]

You use that same process with $sin \theta = \frac{y}{r}$

[ocean seal]

dldh06

[wary stream]

You know theta, the angle

And r

Plug those in

Solve for y

[orchid bison]

is he looking for y or theta?

[alpine sable]

y

and x

okay so 46 = y/5?

[wary stream]

No

You're using sin

The trig function

[lone heart]

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[alpine sable]

Is that not 46 = y / 5

[wary stream]

No it's not that

[alpine sable]

angle = y / r

sin = y / r

[wary stream]

It's $\sin \theta$

[ocean seal]

dldh06

[wary stream]

Where theta is the angle

You don't replace the sine

[alpine sable]

I thought sin theta just literally translates to y/r

like that is the function body

just y/r

[wary stream]

Theta is a variable

[wary stream]

Sine is a function

[orchid bison]

sin is y

x, y = cos, sin

[alpine sable]

I dont see what the word sin is adding

[wary stream]

It's a trig function

[alpine sable]

and the function definition is just y/r

[orchid bison]

remember soh cah toa

[wary stream]

But sin theta is a value

[alpine sable]

the value is not used in the function definition

[wary stream]

,w sin pi

[ocean seal]

Results provided by WolframAlpha

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Upgrade to WolframAlpha Pro!

Requested by dldh06#2659

[wary stream]

$\sin \pi = 0$

Not pi = 0

[ocean seal]

dldh06

[wary stream]

It is a function, like how y = 2x

You don't magically get rid of the x

And say y = 2

Same thing with sin

[orchid bison]

like this

[wary stream]

It's part it

[orchid bison]

its like a slope/ function

[alpine sable]

when I see functions like f(x) = 2x

the input is being used in the function

x appears in the function body

[wary stream]

Same thing with sin x

Or sin theta

[alpine sable]

where is theta in sin(theta) = y/r

[wary stream]

It's in the definition

[alpine sable]

The definition is y/r

[wary stream]

Saying just sine means nothing

Sine takes in an input value

And returns an output

[alpine sable]

I don't see the input value being used anywhere in the function body

[wary stream]

Hence $\sin \theta$

[ocean seal]

dldh06

[orchid bison]

sine (theta) is like y of (angle)

[wary stream]

It is defined as $\sin \theta =\frac{y}{r}$

[ocean seal]

dldh06

[alpine sable]

Do you see how in f(x) = 2x, x is appearing twice?

once in the function signature, the other in its definition?

I only see theta appearing once

in the signature

not in the definition

[wary stream]

What about y = 2x?

It's the thing as f(x) = 2x

[alpine sable]

the input x is just implied there, the signature is replaced with the output

[wary stream]

But x appears once

You need to review trig functions
Here is a short video
https://www.youtube.com/watch?v=WvoFgL4P_rw&ab_channel=TheOrganicChemistryTutor

[alpine sable]

it's fine for it to appear once it it's in the actual mechanics of the function

if I sae y = theta/r, I'd be fine

[wary stream]

Even in the image, it's $\sin \theta =\frac{y}{r}$

[ocean seal]

dldh06

[alpine sable]

Yeah im questioning the image

f(x) = y/r is essentially what I'm seeing

[wary stream]

For trig functions, it's defined including the theta, meaning for example, $\sin \theta = \frac{y}{r}$

[ocean seal]

dldh06

[wary stream]

Meaning if you input an angle of 45 degrees

Meaning $\sin 45$

[ocean seal]

dldh06

[wary stream]

You will return the y coordinate at 45 degrees

Meaning $\sin 45 = \frac{\sqrt{2}}{2}$

[ocean seal]

dldh06

[wary stream]

You can question all you want. Overall, you are not understanding the concept of trig functions properly

[alpine sable]

all I know is that sin is a ratio of two sides. I don't get how it works if we know nothing about the sides

[orchid bison]

this only applies to a unit circle right?

[wary stream]

Yes, that example

[alpine sable]

other than it's magnitude

[wary stream]

You know r, and theta

Meaning you can plug in theta

Into the equation I gave

$\sin \theta = \frac{y}{r}$

[alpine sable]

the equation being sin theta = y / r

[ocean seal]

dldh06

[wary stream]

Yes

[alpine sable]

sin 46 = y/5

[wary stream]

Yes

Solve for y

[alpine sable]

y = 230?

[wary stream]

How?

[orchid bison]

i dont think thats y

[wary stream]

What did you do to get that value?

[alpine sable]

Multiplied both sides by 5

46 * 5 = 230

Y/5 * 5 = y

[wary stream]

As I stated, it is not just 46

It is sin 46

[orchid bison]

here ill write it for you

[wary stream]

Literally this statement here

[alpine sable]

what am I supposed to do with that word

sin

I dont get what it's doing in the equation

[wary stream]

It's a function

[alpine sable]

it's just the name

of the function

[wary stream]

It's a trig function

What about ln?

[wary stream]

If you have ln 5 = y/5, what would you do?

Ignore that ln?

[alpine sable]

I dont know what ln is

[alpine sable]

$sin(45)=\frac{1}{\sqrt{2}}$

U multiplied it with the magnitude?

[wary stream]

Natural log

[ocean seal]

Pi Creature

[wary stream]

No, unit circle

[wary stream]

Do you know log?

[alpine sable]

I dont

[wary stream]

Like $\log_{10}2$

[ocean seal]

dldh06

[alpine sable]

Yeah I haven't learned logarithms yet

[dawn birch]

have u learned exponentials

like 10^x

[alpine sable]

Yes

[dawn birch]

logarithms are the inverse of exponentials

so $$10^{\log_{10}(x)} = x$$

[ocean seal]

jnmwn

[dawn birch]

similarly, $$\log_{9}\left(9^x\right) = x$$

[ocean seal]

jnmwn

[dawn birch]

you can replace the number with any other

[alpine sable]

$sin(\theta)=\frac{y}{r}$

Uk $\theta$=46 degrees and u already know $r$ so just create an equation for y

[dawn birch]

besides like 1 and 0 because they’re weird

[alpine sable]

Where did logarithms came here

[wary stream]

It's because they "remove" the sin

[ocean seal]

Pi Creature

[wary stream]

And just do 46 = y/5

[dawn birch]

so log is to exponentials like square root is to squaring essentially

[alpine sable]

Ah i see

By they u mean m right?

can you show me the equation

I have no idea how I'm supposed to handle the word "sin"

in the equatio

[alpine sable]

Try it yourself at least

[wary stream]

So logs came into play because logs are part of the definition, like how sin is part of the calculation

[orchid bison]

sin is y

[alpine sable]

Uts not a word

[wary stream]

Yes

[alpine sable]

I've tried three times now

It's a function

[wary stream]

Sin is a function

It's a button on the calculator

[last ether]

What's a button?

[alpine sable]

wait so when I read sin(angle) = y/r

[last ether]

Well you wanna isolate y, correct

[alpine sable]

are you saying that sin(angle)'s internal logic, formula whatever

is not y/r

[last ether]

No

Well

sin(θ) = y/r

The whole sin(θ) is represented by y/r

[alpine sable]

is the statement sin(angle) == y/r or sin(angle) = y/r

[wary stream]

No, the calculator, uses taylor series, to find the value

[alpine sable]

OK so that is why I'm confused

[last ether]

Yeah no it would have to like memorize a fuck ton then

[wary stream]

Overall the concept is $\sin \theta = \frac{y}{r}$

[ocean seal]

dldh06

[wary stream]

Where theta is the angle, y is the y value/coordinate, and r is the radius

[alpine sable]

I've been interpreting it like this if anyone programs

function sin(angle) {
  return y/r
}

but you're saying the definition of sin

is hidden away

some big Taylor series thing

OK so then

is it possible to get this point

without using a calculator

[wary stream]

Not easily

[alpine sable]

/ depending on a super complicated Taylor series formula

[last ether]

Tbh, just keep sin(46) as sin(46)

[wary stream]

If you wanted to use trig functions without a calculator, the easiest values would be the ones that line up with the unit circle

[last ether]

It's mathematically equivalent to the actual value of sin(46), being that sin(46) = sin(46)

[tribal heart]

Yes

[alpine sable]

yeah im just wanting to be able to have it all visually click

[last ether]

Wait are you finding the (x, y)?

[alpine sable]

so that when I'm doing game development and I use cos(angle) to get the x position of a point, I understand why

but now it's sounding like it's going to continue being a black box to me until I learn Taylor series

that I have to just be satisfied with "sine and cosine are magical black boxes that have complicated mechanics happening under the hood"

[tribal heart]

Doesn’t the game engine have a built in calculator

[last ether]

Oh wait that's just $(r\cos{(θ)}, r\sin{(θ)})$

[alpine sable]

yes they have sine and cosine functions built in but I dont understand how they work internally

[ocean seal]

Umbraleviathan

[tribal heart]

Wdym work internally

[wary stream]

It's internally, how calculator does it. Because the calculator doesn't store x and y values into the memory

[last ether]

Taylor/Maclaurin series. Stuff you shouldn't work too much

[tribal heart]

You just need the points

[last ether]

But if you really wanna know

[alpine sable]

I have no idea how sine() gives an answer

[wary stream]

It's how calculators determine the value of sine

[alpine sable]

using only angle as the information

[last ether]

There

[tribal heart]

Yes

[last ether]

That's what the calculator uses

[last ether]

@alpine sable

Use your polar identities

x = rcos(θ)
y = rsin(θ)

[alpine sable]

yeah I know that works

I just don't visually get it

when I subtract vectors, it clicks for me. I understand why the result is occuring

not here though

[tribal heart]

It’s just unit circle

[alpine sable]

the sin and cosine are magical black boxes

[wary stream]

So for example, $\sin 45 = \frac{\sqrt{2}}{2}$. That uses the concept of the unit circle and returns the y coordinate

[ocean seal]

dldh06

[last ether]

Well actually I think it's easier to show them this

[alpine sable]

Literally the entire server is here for a simple trig problem

[last ether]

sine is y/r

r(y/r) = y

[last ether]

💀

Use your trig ratios

sine is y/r

Multiply that by r

You get y

[wary stream]

Because the calculator does not store the unit circle in memory, because that's based on set coordinates, the calculator uses Taylor series to calculate the values

[last ether]

@alpine sable does this visual and explanation help?

[alpine sable]

These two

[wary stream]

Because you can calculate sin 46 if you wanted to, but using the unit circle, it is not on there

[tribal heart]

Where do these formulas come from

I’m curious

[wary stream]

It's Taylor series

[alpine sable]

Have u learned calculus yet

[tribal heart]

No

[alpine sable]

It's taught in calc 2

[tribal heart]

🤡

[alpine sable]

Generally ig

[alpine sable]

Wdym sine is y/r

[wary stream]

If you wanted sin 1.234123, the calculator will use Taylor series to find the y coordinate

[alpine sable]

I thought we established that sine is not y/r, but is just equal to it

[alpine sable]

There are btw some trig identities u can use to calculate many angles without Taylor series

[last ether]

The ratio

Here let me show you

[tribal heart]

Y value over radius of circle

[last ether]

You know that sine is the ratio of opposite/hypotenuse, correct?

[alpine sable]

Ig M is still thinking sin is a word

I'm trying to distinguish between the output of a function being equal to some formula, versus the actual definition of a function being equal to some formula

[tribal heart]

Opposite is y value and hypotenuse is r

[alpine sable]

Rather than a function

is anyone here a progeammer

[wary stream]

Yes this

[tribal heart]

Yes

[alpine sable]

Me

Programming for 5 years not professional tho my hobby

[last ether]

Why would you wanna understand the programming behind it? I'm trying to show you why x = rcos(θ) but programming isn't gonna help

[tribal heart]

Ikr

[last ether]

Rather, I can show you the geometry behind it

And the algebra

[alpine sable]

OK, when you guys say "sine is y/r", I'm interpreting that as "the definition of the sine function, is y/r"

I'm interpreting it like this

function sin(angle) {
  return y/r
} 

when it sounds like what you guys actually mean, is that the output of sine(angle) is == to y/r

[last ether]

Yeah

[alpine sable]

Ya

[last ether]

sin(θ) = y/r

[tribal heart]

Yes

[last ether]

Tf is a long double

[sweet wren]

yo

[last ether]

Sounds like something funky grunky

[sweet wren]

what do u program

[alpine sable]

import numpy as np
np.sin(46/180*np.pi) 

okay so you guys mean sine == y/r, not sine = y/r

yes?

[sweet wren]

what do u pogram

[tribal heart]

That is python I think

[alpine sable]

Ok so what's the difference between == and = according to u

== only makes sense for booleans

[alpine sable]

Equivalence versus identity

[wary stream]

If you're asking about your channel, don't try to bump your question

[alpine sable]

I mean the output of it is a bool

[sweet wren]

wdym

[wary stream]

https://www.tutorialspoint.com/cplusplus-program-to-calculate-the-value-of-sin-x-and-cos-x

[alpine sable]

I am into ML

[alpine sable]

He is using javascript

Or maybe she

To say f(x) = 2x, is to say that when you plop in an input into f(x), the internal mechanics that happen to it are 2x

[sweet wren]

or maybe deez

[alpine sable]

to say f(x) == whatever, is to say that the output of f(x) will coincide with whatever

[tribal heart]

Everyone, please type a large "Ha Ha!" in chat to appreciate the phenomenal wit shown in this extraordinary display of the highest form of humor

[last ether]

This is great

This is a moment

[alpine sable]

See

import numpy as np
theta=46
sin_theta=np.sin(theta/180*np.pi) 

Done

f(x) = 2x ---- definition of a function

We can then use that function
f(2) == 4

[sweet wren]

you okay bro

[slim escarp]

Help my boy e-10 out #help-1

[last ether]

Yeah but sine is a special function

[sweet wren]

you cannot be talking if you havent reached e-10 level of math

[last ether]

Wrong reply moment

[wary stream]

You're trying to relate trig with code, which is overall confusing you. The general concept is $\sin \theta = \frac{y}{r}$ where you input a value for theta, and it returns a y coordinate value, with the radius of 1

[ocean seal]

dldh06

[alpine sable]

I want to understand the internal mechanics of how it's able to return a y coordinate using only an angle

[sweet wren]

you cannot understand if u do understand

[alpine sable]

For that u need to learn Taylor series

How can you get a y using just an angle

[tribal heart]

It doesn’t rly matter what formula the calculator uses for sin as long as it works

[last ether]

and I've been trying to show you

[alpine sable]

this is how computers compute it

[last ether]

without code

[sweet wren]

you guys got it all wrong

[wary stream]

Because $\sin \theta = \frac{y}{r} \rightarrow r \cdot \sin \theta = y$

[tribal heart]

[alpine sable]

@alpine sable

[last ether]

Bro this channel a circus fr

[alpine sable]

Too many cooks

=ruin the.....

[last ether]

I thought it would be easier to show the geometry

Like

[tribal heart]

[sweet wren]

its really not that hard to explain

[ocean seal]

dldh06

[last ether]

You know that sin(θ) is y/r

[tribal heart]

Ikr

[sweet wren]

your making the person more confused

[alpine sable]

Depends on who ur explaining to

[wary stream]

No cares what you think

[tribal heart]

There are 5th graders who understand this stuff

[alpine sable]

True

[tribal heart]

Why do you overcomplicate

[alpine sable]

I learned in 6th grade

[last ether]

$r\sin{(θ)} = r\cdot\frac{y}{r} = y$

[alpine sable]

The basics ig

[tribal heart]

Same

[ocean seal]

Umbraleviathan

[last ether]

@alpine sable

There

[alpine sable]

Ok too many cooks

[last ether]

That is the "internals" of it

Geometry, and algebra

[alpine sable]

Are basically ruining the dish

[tribal heart]

Lol

[alpine sable]

So one cook says bye

[last ether]

No code, no series

[wary stream]

Because here
#help-0 message

[last ether]

And then the pot fucking exploded and caused a nuclear blowout and New Jersey took over the world

[tribal heart]

It’s not that hard

[alpine sable]

True

If this amount of trig is hard then trig problems would be like....

[last ether]

Pain and suffering

[tribal heart]

Wdym amount of trig is hard

[alpine sable]

I mean just understanding what sin is

[tribal heart]

Ah yes

[wary stream]

The OP, is assuming that "sin" is just a word and not understanding that sin is a function that takes the input of the angle and returns the y coordinate

[tribal heart]

Haha

[alpine sable]

Where do we go in this photo

What's the first step

[tribal heart]

There was a funny amc12 problem I saw on trig

[wary stream]

Label

[tribal heart]

It had both sin and cos be 1

[wary stream]

You know r and the angle

[alpine sable]

I've always understood what sine is a function, my confusion has been that I've been interpreting sine = y/r to be the function definition rather than just an equivalence statement

[tribal heart]

My brain is fried

[wary stream]

Not sine = y/r but sine theta = y/r

[alpine sable]

Do u understand sin and cos properly no?
Open this and read https://www.mathsisfun.com/algebra/trigonometry.html

That's all i can say

[wary stream]

In classes, it's been taught as sine = y/r because the theta is omitted

[last ether]

Let us look at this like a graph

If this was a right triangle

[wary stream]

In actuality, it's defined as sine theta = y/r

[alpine sable]

yes I've watched a billion videos about sobcahtoa. I understand that sine is equal to the ratio of opposite over hypotenuse

[last ether]

Well here

Lemmme like

[alpine sable]

Then what's ur doubt about it?

How it relates to angles

Then read that doc it's clearly stated

Using a unit circle

We use a unit circle

[wary stream]

https://www.youtube.com/watch?v=PUB0TaZ7bhA&ab_channel=TheOrganicChemistryTutor

[alpine sable]

all that tells us is that r is 1

[wary stream]

Here watch that

[alpine sable]

No!

Read it

U will get it

[orchid bison]

it seems like M- doesnt know the basics

[wary stream]

That is based only on the unit circle

R can be anything you want

It's scaled

[orchid bison]

a circle with 1 as the radius is called a unit circle

[last ether]

That's where the x and y comes from

[wary stream]

Getting into the weeds, it's a scaling factor

[tribal heart]

This is a clown fest

[wary stream]

The general concept where r = 1, is used with the unit circle

[alpine sable]

OK so we have this same scenario

But instead of 5, r =1

[wary stream]

If r = 3, the coordinates get scaled too

[alpine sable]

r = 1

[orchid bison]

thats r = 5

the hypotenuse is the radius of the circle

[alpine sable]

I'm saying if it were the unit circle

R would be 1

[orchid bison]

yep

[alpine sable]

Okay so let's pretend that's how it is

r = 1 angle = 46

[orchid bison]

yes

[wary stream]

You can't not with your problem

Because r = 5

[alpine sable]

I'm trying to understand how sine works internally with the unit circle first

Scaling it with r later makes sense

[tribal heart]

smh

[wary stream]

If r = 1 then sin(46) = 0.719339

[alpine sable]

Okay how do we calculate that

[wary stream]

Using a calculator

[last ether]

Calculator. Just keep it as sin(46)

It's mathematically equivalent

sin(46) is a number in itself

It's irrational, but it's a number

[alpine sable]

but someone earlier said that you could understand the internal mechanics without needing to know the Taylor series

[last ether]

conceptually you can understand it

[alpine sable]

Ok so summarizing the entire thing first fact forget $sin(\theta)$ and $cos(\theta)$ as ratios of a triangle. Think of the unit circle u have walked a particular arc length in that unit circle the let's say the length u walked is $\theta$ this arc length is called radians. The point u end up with is $(cos(\theta), sin(\theta)$ . Now u strech the hypotenuse or scale it by a factor 5 so the x and y coordinate gets streched by a factor of 5 hence new point becomes $(5cos(\theta), 5sin(\theta))$

[last ether]

There's a major difference

[alpine sable]

and i hope you know the relation between arc length and angle

This is all

[last ether]

Bro you got nice colors, I gotta deal with piss green and spoiled brown

[orchid bison]

thanks

[ocean seal]

Pi Creature

[alpine sable]

Here theta is equal 46 degrees so the arc length is 46/180*π

[orchid bison]

nvm its the same

[alpine sable]

U can calculate sin(46), cos(46) by using something called a Taylor series which will help u approximate it by hand.

But generally

Use a calculator

Now bye

[wary stream]

It's getting into the weeds, but as mentioned, the calculator does not store the unit circle in memory, because that's just a waste of space, because between 0 and 360 degrees, there are infinitely many values. So instead of storing in on infinite amount of values and just taking up tons of memory, the calculator uses a concept, called Taylor series, and sin has a specific Taylor series formula. So instead of storing infinitely many values between 0 and 360, the calculator just needs to store one formula, has a decent amount of terms to get the accuracy high enough, to calculate sin values

[last ether]

I wanna buy your new rap LP

[orchid bison]

im gonna go

[wary stream]

And to mention, trig functions are periodic, meaning every 360 degrees, it repeats, so the value at 360 is the same as 720, and so on

[orchid bison]

good luck on this

[wary stream]

So internally, the calculator stores, these equations, to do trig, where all you do is plug in a value for x and it spits out a value, using that equation, because like I said, it's easier and takes less memory to store that equation than infinitely many values

Because you can calculate sin 1.2, if you wanted to

But I'm going too, so good bye

Look into trig and ask your teacher, that's all I got for you now

[last ether]

(I have a simplified proof if your brain hurts)

(Just ping or DM me)

(Also geometry dash crashed and I just lost 15 minutes of work ffs)

Ope he dead

[alpine sable]

I thought that trig stuff dates back to ancient Greece before we had calculus

why is every explanation I'm seeing for sin/cosine being these giant derivative calculus stuff equations

[last ether]

No calculus is needed

Lemmme show you a simplified proof

The radius has a slope, correct?

[alpine sable]

Yes

[last ether]

So imagine a circle on the x, y plane

Centered at 0, 0

And the radius is 1, to make things easier

The slope of that radius is y/x (rise/run)

You might also notice you can construct a triangle

So the GREEN is the radius
The shitty-drawn black line is the angle, θ

You get me?

[alpine sable]

Yes, but we don't have y/x yet. All we know is r=1 and angle=45

[last ether]

Yes, be patient though

Because shit gets real

[last ether]

When you construct a right triangle, this happens:

The opposite leg becomes Y
The adjacent leg becomes X
The hypotenuse is the radius, R
And the angle between X and R is θ

Do you understand that?

[alpine sable]

Yes

[last ether]

Now let us look at finding what the X coordinate could be

We can understand that from this triangle, the cos(θ) = X/R

Do you understand why?

[alpine sable]

I dont

[last ether]

Trigonometry identities. Adjacent/hypotenuse

[alpine sable]

Okay but

this is begging the question

you can't reference the sine/cosine functions before we have established it

I have no idea what cosine(theta) is doing

[last ether]

It's doing this

That's what cos(θ) is doing

It's essentially taking the radius and forming a right triangle

And then taking the adjacent/hypotenuse of that triangle, for whatever θ is

[alpine sable]

How is it doing that?

[last ether]

Well let us think about it

The green is the radius

So

[wary stream]

I still believe that you should just go to your math teacher and spend time talking to them about this

[last ether]

If you make a right triangle

It's turning it's slope into a triangle

[last ether]

Geometry, basically

[alpine sable]

more internal black box magic

[last ether]

Not really

It's just making a triangle

[alpine sable]

How is it doing that?

[last ether]

That's just how it works

Geometrically

And because that's the definition of cosine

[alpine sable]

what is the definition of cosine

it can't be this ratio

[last ether]

The ratio of the adjacent side to the hypotenuse

[alpine sable]

because clearly you're saying the output of cosine is equal to that ratio

[last ether]

It is

[alpine sable]

so that is not the definition

it's just an equivalence

[last ether]

I think you're getting your terminology confused

The equivalence is the definition

Mathematical definition

[alpine sable]

Mathematics apparently makes no distinction between function definitions and equivalence

and that's confusing me

[last ether]

Equivalence is the mathematical definition

Not the dictionary definition

[alpine sable]

saying that the output of a function is equal to some formula

[last ether]

You understand that we can construct a right triangle with these sides

[alpine sable]

you can't say that the formula is the definition

or how would there be an output

To begin with

[last ether]

This isn't about coding

This is about conceptualization

[alpine sable]

I conceptually do not understand what sine(angle) could possibly be doing internally to return an output

if we apparently have no internal definition of it

except some equivalent relationship between its output and some ratio

[last ether]

The sine is the ratio of a supposed opposite side to its respective hypotenuse, in relation to an angle

That is its definition and formula

Formulas do not need to be things like f(x)

Internally, it's that whenever we construct a specific angle, there will always be a ratio ascribed to it given these conditions

So when we have an angle of 45° with a radius, we are able to go backwards and set ourselves the conditions

We know that the radius, of some length, has a definition height and a definite width

[alpine sable]

how do we get the height and width

[last ether]

The height, we know as the Y; the width, the X

Because think about the X, Y graph

X goes left to right, Y goes bottom to top

Width; height

This is, of course, conceptual

But you might realize that every line segment has a definition X-width and Y-height

The radius, being a line segment, is characterized (internally) as having width X and height Y

When we draw that out, we get a right triangle

Hence this drawing

[alpine sable]

Yes

[last ether]

So that is where we get the triangle

Now because of this internal characterization

We can "dissect" it and look at it from all parts

We know that the radius has some value of Y

And some value of X

These X and Y values are, as previously mentioned, ascribed to their respective angle

In order to get these, we have to look back at which trig functions have X an Y in them, given these conditions

sin(θ) = Y/R by trigonometric definition (radius height/radius length)

cos(θ) = X/R by trigonometric definition (radius width/radius length)

So where is it internally? It's built innately, by geometry

But now we gotta get rid of the R's in the denominator. Which is easy: just multiply by R

Rsin(θ) = (RY)/R = Y

Rcos(θ) = (XY)/R = X

The real definition of these trig functions is the ratio of innate lengths should the angle be a part of a right triangle with unit (length of 1) hypotenuse.

Which answers your question about the ratio being the "definition" of cosine. It is the mathematical definition. But it's multivariable.

[alpine sable]

The beginning of this, we set out to determine an x,y point knowing only a radius and an angle.

In your demonstration of doing this, you have just stipulated a given x,y coordinate and then appealed to the sin/cosine functions to explain how we get them

when I'm still lost on how sin cosine get us it with only the angle and raidus information

just feels like an infinite regression

[last ether]

It's built within the innate characteristics of sine and cosine

For whatever how the Greeks determine it. That's beyond my ken

And rather, unimportant

I'm sure someone with a degree can explain the origins of it

[last ether]

Well yeah, you do have a given (x, y), but it's not readily defined

It's at the end of your vector

You just don't know what (x, y) is, but you know it's there

Which is why they remain in variable form

These are good questions, ngl. And perhaps your teacher can explain it better.

It's best to understand that the Greeks had this pre-set determination that for any right triangle with a definite width and definite height and a hypotenuse that connects the two will always form a definite angle.

They characterized this definite triangle by its definite height, Y, and definite width, X, on the XY plane

For every "X" and "Y", whatever they may be, they found out that those with the same ratios formed the same angle

Which is why it's innate

The (x, y), apart from being the tail of your vector, has been defined into the very conceptual definition of cosine and sine

However, this also means that the radius length (hypotenuse) is definitely defined

---- so here's your answer: ---

In your case, it is 5

And it's hoisted at a 45° angle

So it will have some definite width and some definite height to it, by the converse of the layman Greek trigonometric definitions.

We know that the sine is height/length (opposite/hypotenuse), but perhaps we can manipulate that to our benefit without changing the definition

sin(θ) = Y/R

And if you do simple algebra and wanna isolate Y:

Rsin(θ) = Y

@alpine sable does this make more sense

TL;DR the Greeks were so fucking smart that they found out that any definite right triangle with the same side length ratios will have the angle, and vise versa

I can't wait for the other two guys who were here to show up and be like "bro what the fuck"

[lone heart]

@alpine sable Has your question been resolved?

[alpine sable]

A little bit yeah, I need to study it more. Thanks for your help

[lone heart]

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[woeful dock]

hello i need help knowing if i’m doing this correctly or not, i’m so lost

[gray isle]

the initial approach is valid though a bit inefficient
what do you have after dividing both sides of
-4k < 0
by -4
and how did that lead to k=0

[woeful dock]

would it be k<0

[gray isle]

no

is -4 a positive or negative number?

[woeful dock]

negative

OHH i have to switch the symbol right

k>0?

[gray isle]

yes

if k > 0, there will be no real solutions
and it seems they only want 1 such value

also this conclusion would be quite intuitive from simply subtracting k from both sides of the initial equation to get
(x+3)^2 = -k

[woeful dock]

so can i choose any integer greater than 0 as the answer?

[gray isle]

based on the wording of the question, yes

[woeful dock]

okay, thank you!

[lone heart]

@woeful dock Has your question been resolved?

[woeful dock]

yes

[lone heart]

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Remember:
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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[spring jackal]

how do i convert (x-1)/(2-x)=2x+1 into ax^2 + bx + c = 0 form?

[tacit arch]

Multiply both sides of the equation by the denominator

a/b = c if and only if a = bc

[spring jackal]

alright so (x-1)/(2-x) x (2-x) = 2x+1(2-x)?

[tacit arch]

Need parentheses, but sure

[spring jackal]

• (2-x^2)/(2-x) = x+2?

[gray isle]

how are you getting that

[spring jackal]

online calculator although maybe i put it in wrong

yeah i definitely put it in wrong im not sure the correct way to input it

[gray isle]

the correct way is don't
and do it yourself by hand

[spring jackal]

rhs = -2x^2+3x+2 im not sure how to do the lhs by hand

[gray isle]

consider why you're even multiply both sides by (2-x)

[spring jackal]

hold up its x-1

[spring jackal]

to get rid of the fraction right?

[gray isle]

yes

[spring jackal]

alright i got the answer correct, thank you

.close

[lone heart]

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Remember:
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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[cinder sundial]

Does it always works?

I know if I assume alpha=45 and beta =35 then the equation is works

But I’m not sure if it’s work if the value of alpha, beta changes

And I can’t seem to figure it out myself

[gloomy wolf]

let alpha = beta

[lethal stump]

yea it don work

[cinder sundial]

Ohh if the equation works under those 3 circumstances 1.alpha>beta, 2.alpha<beta, 3.alpha=beta

Then it’s always works I guess

[cinder sundial]

Could you explain why

[lethal stump]

wait are you trying to find sols to this eqn?

or are you trying to prove that it works

oh wait nvm i reread ur question

[gloomy wolf]

$\frac{1}{tan(\alpha - \beta)} = \frac{\cos(\alpha - \beta)}{\sin(\alpha - \beta)}$

[ocean seal]

JamesH

[gloomy wolf]

when $\alpha = \beta$, this is undefined

[ocean seal]

JamesH

[cinder sundial]

“Undefined” I’m not sure what does it mean in math

[lethal stump]

if you expand using angle sum formulae you'll see that it doesn't always work

[gloomy wolf]

it means whatever $\frac{1}{0}$ means

undefined

[ocean seal]

JamesH

[cinder sundial]

Got it

.close

[lone heart]

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Closed by @cinder sundial

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Send your question here to claim the channel.

Remember:
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[cinder sundial]

given that tan(alpha)+tan(beta)=5/2, tan(alpha)tan(beta)=3/2.
0<alpha<beta<90degree, then the question asks for tan(alpha-beta)

[cinder sundial]

i have no idea about this one

[vale wigeon]

let u = tan(alpha) and v = tan(beta), then you're given u+v=5/2 and uv=3/2

[gloomy wolf]

i suggest you rewrite as sines and cosines

[vale wigeon]

so u and v are the solutions of of 2x^2 - 5x + 3 = 0

[gloomy wolf]

oh also this

[cinder sundial]

yes, ure right

[vale wigeon]

this way you will get tan(alpha) and tan(beta) individually and you can use the angle difference formula for tan to get what you want

[cinder sundial]

yes ure right

let me try

[lone heart]

@cinder sundial Has your question been resolved?

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[lament glen]

saw a comment in a youtube video which made me wonder,

[lament glen]

I mean the floor function seems to create 1 wide rectangles

[formal tree]

Yes, they have the same value (infinity).

[lament glen]

ah aight

.close

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