geometry-and-trigonometry

[dense bough]

of course it exists before you learn calculus

but when you meet the edge cases it pokes holes in the concept of a 'tangent'

[grave marsh]

look im telling. there is no line passing through 0,0 which is not the x-axis which intersects the parabola x^2 exactly once

[grave marsh]

okay

ah okay i thought it was x^2

but still it holds yes

oh yes you're correct, there's no exact "tangent" then

yeah

the direction becomes tough for light to choose lol

[unkempt glen]

this took WAY too long: https://www.geogebra.org/3d/xcjmq6hg

any quadrilateral is a perspective image of a square (drag ABCD around and you can always produce a square image)

[elfin vine]

guys if i wanna solve qs using the form of general solution given below how should i proceed from the given step above

[pure cape]

please ping after 15min

and just imagine x-pi/6 as a whole

so let beta be x-pi/6, alpha be pi/3

you have cos(beta)=cos(alpha)

solve for beta in terms of alpha

[elfin vine]

ok thanks

[elfin vine]

sorry bout that

[elfin vine]

I need to find the general solutions for the following question but I can't seem to find the tan alpha
help pls ?

[elfin vine]

help ?

[elfin vine]

pls put this stuff in #chill

[obtuse patio]

why did i get pinged

did someone ping helper

@jagged kindlestop with the spam

[obtuse patio]

stop

[jagged kindle]

ok 👍

[obtuse patio]

ayo @forest dove are@jagged kindle 's messages spam?

they posted in like many channels

[jagged kindle]

you are dumb! haha!!!

[obtuse patio]

8 channels

wut

why cant i post that

its an image of post history

[elfin vine]

<@&286206848099549185>

pls help

[obtuse patio]

okay so

all u do

is do arctan of both sides

find alpha

the two solutions are

alpha and 180+alpha

[elfin vine]

idk how to do that man

[obtuse patio]

?

tan inverse

[elfin vine]

oh

yea

ik that

[obtuse patio]

arctan = taninverse

just diff names

[elfin vine]

thats new to me thanks

[obtuse patio]

gotcha

np

[unkempt venture]

task 1 solutions required

https://www.desmos.com/calculator/mz36je6kjx

[obtuse patio]

*guidance

[unkempt venture]

ok guidance

[obtuse patio]

this is an annoying problem i gotta say

u know calc?

[unkempt venture]

ye

[obtuse patio]

why post in geo then

[unkempt venture]

I am a beginner

so not sure

walk ne through it if u can

[obtuse patio]

calc > geo

geo is like 4 years before

w/e+

so

[unkempt venture]

cool

[obtuse patio]

first one: a(t) = 0.5, a'(t) < 0, b(t) = 3/4 * A, b'(t) > 0,

does that help

[unkempt venture]

Differentiate?

a' = 0 it's a constant

[obtuse patio]

no?

its a perioidic function

[unkempt venture]

at = 2sin(2pi/21(x-2.3) + 2

at ' < 0

?

[obtuse patio]

well so

the derivative of sin

is cos

cos goes between pos and neg

there ofc will be a' < 0

[unkempt venture]

and how do you find times as mentioned by derivatives

[obtuse patio]

wdym

[unkempt venture]

I am not sure about the use of derivatives to answer task 1

[obtuse patio]

like if youre allowed to or how

[unkempt venture]

how

[subtle palm]

bro aslong as you get the answer

you fineee

[unkempt venture]

how do u get the answer lol

[subtle palm]

by getting help

oh well time's up gotta wheeeee to precalc

[unkempt venture]

ok forget task 1

what about task 2

its easier

[subtle palm]

sureeee why not

[subtle palm]

look at the intersections in graph

[subtle palm]

dis boi

[unkempt venture]

https://www.desmos.com/calculator/mz36je6kjx

intersections at 0.2?

[subtle palm]

oh wait this involves AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA precisely on y=0.2

[unkempt venture]

I see

[subtle palm]

hngggh-

[unkempt venture]

so all the x value at y = 0.2 are the times

we need

[subtle palm]

yeeeeees

we need to see where it loops to cut corners

opening link...

[unkempt venture]

👍

[subtle palm]

every 42, i think

yep

14x3 and 21x2

[unkempt venture]

Oh so all x values until 42

[subtle palm]

no

it loops after 42

so it becomes easier to locate

[unkempt venture]

k

[subtle palm]

oh wait this isn't geogebra i cant put intersections

wait a sec k

[unkempt venture]

...

I just a little confused lol

it appears to be blinking

[subtle palm]

OH WAIT I READ THIS WRONG

[unkempt venture]

blinking I mean

[subtle palm]

[unkempt venture]

No I read this wrong

[subtle palm]

no i

[unkempt venture]

u shall provide me with the correct solution

[subtle palm]

working on it

[unkempt venture]

👍

[subtle palm]

got the $h(x)=0.2$ function going

[somber coyote]

CalvinTGB11

[subtle palm]

to mark what is blinking and what is not

also might do $i(y)=42$ too

[somber coyote]

CalvinTGB11

[unkempt venture]

ok

[subtle palm]

bruh most of these are non-terminating

[unkempt venture]

hmm

[subtle palm]

u there?

[unkempt venture]

ye

[subtle palm]

yea

[unkempt venture]

so no possible solutions?

[subtle palm]

the only non terminating is 36.75

wait and 15.75

i think that's what they meant

[unkempt venture]

what does it mean by non terminating ?

[subtle palm]

i think these all terminate

not sure tho

btw non-terminating means it's decimal digits keep going

such as 1/7 or 7/18

[unkempt venture]

oh

so only 2 solution we have

[subtle palm]

only 36.75 and 15.75 are for-sure terminating

[subtle palm]

there could be others

they aren't non-viable solutions, their decimal digits just keep going

[unkempt venture]

You would use inequalities.

You do 2cos(pi/7(x-2.3))+2≤0.2 and 2sin(2pi/21(x))+2≤0.2 and solve for ranges of x. Because this is a clock, you only have to look at x from 0 to 1440 minutes (which is 24 hours). After 1440 minutes it will repeat.

[subtle palm]

yes and in that repetitiveness it repeats

with the subdivision of every 42 minutes repeating

[unkempt venture]

But how do u find ranges of x

like how do u display your answer

final answer

[subtle palm]

oh wait i used points lol

where they touch 0

back to work

[unkempt venture]

oh lol

[subtle palm]

and then, calvin proceeds to delete the old points he made and proceeds to find new ones

elevator music plays

[unkempt venture]

you got this Calvin 🥳

lol

[subtle palm]

IT MADE THE DECIMAL EVEN WORSE

[unkempt venture]

[subtle palm]

I CANT-

[unkempt venture]

its algd

[subtle palm]

hol'up

sending in another format-

[unkempt venture]

ok

[subtle palm]

DANGIT

[unkempt venture]

wut happened

[subtle palm]

it sent the same image in pdf, i expected text+image

i'm on a computer btw

[unkempt venture]

oh k

[subtle palm]

it aint copyin

aaaa

tldr all of them are vague on if it terminates or not

since it all has 10+ decimal digits

[unkempt venture]

Can we use inequality

[subtle palm]

it needs specific numbers

inequality is basically the same as ranges

[unkempt venture]

ok

[subtle palm]

cant-

FRUSTRATION FILLS MY BODY TO THE BRIMMMMMM

[unkempt venture]

its alright

[subtle palm]

no is not

[unkempt venture]

U tried atleast

So did i

[subtle palm]

it's just- i don't have discord on my phone

so i can't take a shot from my phone

[unkempt venture]

what's in the shot

[subtle palm]

take a photo********

sdas,dkamkw mklad s

awi

asjnsjkaldvbashjbdjvbhjavhjbhjbhjbhjbhjbhjbhbjbhbjbhbhbjhjbhbhj

YES

sssnbahdvbsahjbdhjabjhdkhid nih wqihdu hiwqjoqhjq

uwqbdbduygwhqjbnuygghjbnmghbmuyggh

[unkempt venture]

lol

[subtle palm]

hnyasgdgasdguagud

at the calculator there's a list of coords

and i can correspond the points to the ranges

BUT i can't take a screenshot

as THIS IS A COMPUTER

a e u g h

[unkempt venture]

do u know sniping tool

on windows

[subtle palm]

nop

k e i n e a h n u n g

[unkempt venture]

go to windows search bar and type sniping tool

[subtle palm]

mhm

[unkempt venture]

did it open

[subtle palm]

yes

OH WAIT THERE'S A HORIZONTAL SCROLL

HOW AM I THIS DERANGED

[unkempt venture]

oh yes

[unkempt venture]

nice

[subtle palm]

(a,b)(c,d)(e,f)(g,h)(i,j)

the rest is yours

the 0.2 is because the max y is 0.2 obviously

[unkempt venture]

did u get these values off the graph ?

[subtle palm]

yep

[unkempt venture]

nice

so this our solution

[subtle palm]

may i snip the whole graph?

[unkempt venture]

ye

if u wish to

[subtle palm]

handcopied the functions from the last website

[unkempt venture]

can u share the link

[subtle palm]

geogebra.org/calculator

but it's a blank calculator

[unkempt venture]

is that demos ?

[subtle palm]

wdym

[unkempt venture]

Oh wait nvm

U did a great job tho

I must admit

[subtle palm]

eh it was just some logic and intersect points

this is the laziest approach

[unkempt venture]

but should I round those decimals?

[subtle palm]

y wuld u

saving space is irrelevant in maths

in maths you have to be precise

[unkempt venture]

I have to answer those tasks on a written sheet lol

[subtle palm]

but engineering....

[unkempt venture]

Fair

[subtle palm]

as i heard engineering rounds pi to 3 and euler to 3

[unkempt venture]

damn

[subtle palm]

be precise

[unkempt venture]

I shall

[subtle palm]

don't be like [insert engineering junior]

be like bob

[unkempt venture]

should I just list the coordinates to answer task 2

Like x = ....

[subtle palm]

use this

[unkempt venture]

k

[subtle palm]

and this

[subtle palm]

convert it into this

[unkempt venture]

How would u write (a,b)

[subtle palm]

take the x value of point a

that becomes a

[unkempt venture]

oh

[subtle palm]

take the x value of point b

that becomes b

[unkempt venture]

can I say ( a to b)

[subtle palm]

no that might confuse the teacher

[unkempt venture]

ye

[subtle palm]

(a, b) is the standard format

[unkempt venture]

k

[subtle palm]

btw where are you from

[unkempt venture]

NZ

u?

[subtle palm]

cos in america is probably 3am

oooo ok

i'm from ID

[unkempt venture]

damn nice

[subtle palm]

so that's why ur awake

[unkempt venture]

yup

got a exam tomorrow

[subtle palm]

lemme guess... it's 5:34 there?

[unkempt venture]

7 35

[subtle palm]

oh wait yea

[unkempt venture]

good try

[subtle palm]

cos' our eastern province is 4:43

and i guessed it was 1 hour east

[unkempt venture]

where u abouts in ID

[subtle palm]

a quick look back on the map actually puts you 2 hours after

[unkempt venture]

ye

[subtle palm]

idk sus tho

[unkempt venture]

jokes

[subtle palm]

my province is small enough for you to be able to find me

[unkempt venture]

fair

[subtle palm]

hint: it ain't bali, bali is the hawai'i it gets the blings

[unkempt venture]

hmm

I know where u r i think

[subtle palm]

do u live in the big 3

[unkempt venture]

nope

lol

[subtle palm]

like auck christ or well

[unkempt venture]

otago

[subtle palm]

what?

[unkempt venture]

I guess it's big

[subtle palm]

thought the kiwi bros only live in cities

[unkempt venture]

I live in south

[subtle palm]

the other lands are de jure controlled by sheeps or somethin

[unkempt venture]

nah somewhat is

but it is residential in otago

u should come

[subtle palm]

is otago a province or a city

oh a province

[unkempt venture]

yej

[subtle palm]

it has queenstown and dunedin

[unkempt venture]

oooo

[subtle palm]

btw did i hear you have 0 cases?

[unkempt venture]

u almost located me

[subtle palm]

like null

[unkempt venture]

Ye

j

0 cases

[subtle palm]

queenstown sounds more B I G

so imma guess queenstown

[unkempt venture]

sound more fun tho

queenstown is tourist spot

but yea I am in Dunedin B I G

[subtle palm]

ohhh ok

it's not like i'm gonna track you or anything

[unkempt venture]

[subtle palm]

discord's encryption is overhaul for ip trackers honestly

[unkempt venture]

Ye easy track

[subtle palm]

:SiLeNcE:

[unkempt venture]

j

[subtle palm]

wut

[waxen glen]

can someone explain how to prove this by the power of point?

[silk patio]

Apply power of a point

Call the intersection of AB with the circle X

Call AX=x

Then b^2=x(x+2a)

Define c=x+a

b^2=(c-a)(c+a)

a^2+b^2=c^2

[obtuse patio]

how

[silk patio]

Power of a point

[elfin vine]

can someone help me out w this
the form for finding out tan x = b/a
where a=rcosx and b=rsinx
but here he the solution has portrayed two different things

1. it has been solved using the gen. sol. of sinx = siny
   and
2. the form for finding out tan x has been calculated by dividing a by b
   that is a/b
   i have no clue why the second thing is happening
   or how it is pertaining to the *1st point
   pls help

[elfin vine]

<@&286206848099549185>

someone pls help ?

pls ?

[obtuse patio]

where u stuck?

[elfin vine]

can figure out why the boxed part has occurred

[obtuse patio]

its to find the angle

[elfin vine]

but tan x = b/a

this dude wrote tanx=a/b

and then found x

[obtuse patio]

?

tan is sin/cos

so divide the first by the second

sqrt3 - 1/sqrt3 + 1 = tana

[elfin vine]

hold up

[dark sparrow]

parentheses!!!

[obtuse patio]

i would but they know what im talking about

also im lazy

[elfin vine]

i dont follow this method of finding tanx=b/a here ?

[elfin vine]

yea

[obtuse patio]

;-;

its just a general form

[elfin vine]

ik

😭

[obtuse patio]

tan is always sin/cos

sometimes a is cos and a is sin

just know that sin/cos = tan

dont try to memorize specific methods

[elfin vine]

ik man

mind poof

[elfin vine]

i dont memorize

this thing just did not strike me

thanks

[obtuse patio]

np

[elfin vine]

my insides are eating my brain after such stupidity

[obtuse patio]

youre fine dw about it

we all get lost

[elfin vine]

https://tenor.com/view/tears-sad-cute-kitty-cat-gif-14193216

[rich wolf]

,calc 94 - 61

[somber coyote]

Result:

33

[rich wolf]

QED

[earnest basin]

Can someone help me

[covert swallow]

go ahead

[earnest basin]

Question:
In an oblique triangle 𝑨𝑩𝑪, where 𝒎∠𝑪𝑨𝑩 = 30 degrees, 𝑨𝑩 = 9.3 cm. and 𝑨𝑪 = 4.6 cm. Solve for 𝒎∠𝑨𝑩𝑪 to the nearest degree.@covert swallow

[silent plank]

Start by drawing a diagram

[upper karma]

Can someone clear this-
When we define the trigonometric function on the unit cirlce it's just an extension of the right traingle definition we say that the coordinates of any point on the unit circle are (cos,sin) where angle can only be measured from x axis but for higher angles is this definition not forced how can say that angles should only be measured from x axis what I am trying to say is sin30=1/2 is correct but saying sin30=sin150=1/2 not forced because we said that angles can only be measured from x axis sin 150 degree exists otherwise it's same as measuring sin 30degrees

[obtuse patio]

@upper karma ^

this is very nice how did u get this

did u maek it

[upper karma]

Sorry I made a typo I meant sin 30 and sin150(not sin120) if you only consider the triangles inside the unit cricle there is nothing like 150 degree until we say that angle can only be measured from +x axis

[upper karma]

No I am saying while finding any trigonometric ratio why angle has to be measured from the +x axis(x axis on the right) I am not saying anything about the -ve x axis

[unique bobcat]

I rlly need help with this problem... I've been stuck on it for hours TwT

[acoustic jungle]

are you saying because sin(x)=sin(180-x), we should instead have the convention that the angle should be measured from the x axis, not just the positive x axis?

you can't do that because it's not the same with cos

cos(x) isn't cos(180-x)

[acoustic jungle]

I have learned a bit of vectors, but I don't understand why u dot v = 2? If u and v are unit vectors, how can their dot product be greater than 1?

[umbral snow]

Obviously cosθ = 2

[unique bobcat]

yeah ikr... but it's really written like this. i think it was meant to be 1/2 (coz that would make more sense)

[sly marlin]

yeah since we suspect a mistake in the problem, maybe direct it to your teacher/prof

[umbral snow]

It's an error but I doubt it will matter. I'm more concerned about how this starts with "in the space". Is something cut off?

[acoustic jungle]

i is capitalized so I don't think so.

[unique bobcat]

hmmm i've tried to translate it from french so it might not be too accurate ....

[gentle fog]

Can I pick someone's brain? This may be more of an advanced topic, but so far I've only used trig math. I'm trying to create a template equation for a brachistochrone curve. I figured out as far as graphing it with parametric equations as a coordinate pair. The curve looks complete and differential everywhere, so I would assume there is a formula for it out there somewhere, but I cannot find it. Anyone have any ideas? Please ping or dm, I won't see the response otherwise.
https://www.desmos.com/calculator/jg9jg57hor
This is a link to some explaining I've done, typed out on Desmos graphing calculator.

[wise pawn]

if I understand you correctly, you want to find the path a point takes on a wheel that's rolling while also sliding, right? @gentle fog

you can combine the two easier motions into one single motion pretty easily, if that is what you want

[gentle fog]

That is what I'm looking for, and I want a formula for y as a function of x, no t variable

[rich wolf]

constant velocity?

[wise pawn]

that won't be possible because the curve back tracks and so is not a function

although that's not to say all hope is lost, x^2+y^2=1 is a circle and is not a function but relates x and y

the fact that you have it parametrically already should mean you can in principle solve for t and plug in one to the other to get your formula

[surreal pike]

help

[wise pawn]

something funky happens at the peak, but if you plug in for t and solve like I describe I end up with this:

$y=R\sin\left(\frac{q}{s}\left(x-\sqrt{R^{2}-y^{2}}\right)\right)$

[somber coyote]

Merosity

[gentle fog]

@wise pawn That is very interesting, I'd love to find out how you got that. But strangely, that is only true for every set of 4 x values, ie 2<x<6, 10<x<14, etc

[lofty pendant]

Is the circle a polygon or no?

[dark sparrow]

what is your definition of a polygon?

[solid temple]

Is Homeomorphism same as Coordinate Transformation, what's the Differentiation between the Both?

Anyone know how to go into #point-set-topology server?

[solid temple]

I want to enter, but, don't know How to?

[wise pawn]

I just did this @gentle fog. Also if you graph the same thing with - sqrt you get the other half, it turns out it's graphing both paths of points that are 180 degrees apart

[tepid crow]

#get-advanced-access

Click on the green checkmark reaction

[cloud stump]

Menelaus's theorem

[upper karma]

can someone just confirm if i did this correctly

[livid moss]

What is meant by degrees in standard position? @upper karma

Like measuring from the positive x axis (like East)?

[upper karma]

im not really sure myself

[livid moss]

[upper karma]

only part i'm kinda confused on is the 2nd part

[livid moss]

,w arctan(3) in degrees

[somber coyote]

Results provided by WolframAlpha

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

Requested by Lunasong#7197

[livid moss]

Second one is fine

First one is fine if it means what I think it means

[dark sparrow]

do we know that the blocks are square tho

[livid moss]

I assumed

[upper karma]

okay i tried another one

this 1 also good?

[young bloom]

SBA = 60 degree
ABCD is a square with the length of "a"
SA perpendicular with ABCD
find the length from D to plane SBC

[young bloom]

ok just solved

[pure cape]

hmm just wondering if we can prove DC is perpendicular to SC
have hunch that it is

but i dont think so

[young bloom]

its not

[pure cape]

hmmm right

so then its just the process of drawing a perpendicular line to SC and find it

pretty tedious

[young bloom]

nvm i just solved it

[pure cape]

👍

[young bloom]

SA = sqrt(3) a

SB = 2a

find height of A and done

[pure cape]

tho i have a doubt, i dont think the distance from A to D are the same

since truly the sides are indeed parallel

but the 3d shape is not symmetric

because of SA being perpendicular to the base

hence the height from D to SBC will differ from A to SBC

[young bloom]

it on the same plane

if a line parallel to a line within a plane

it also parallel to the plane itself

[pure cape]

oh wait nvm you're right, AD is parallel to SBC

[young bloom]

is there away to find the parabola with just 3 points?

[versed river]

yes, any 3 points uniquely define a parabola

(so long as they arent colinear)

you know it has equation y=ax^2+bx+c

so then you can plug all your points into that equation and get a system to solve for a, b, and c.

[young bloom]

okay

[versed river]

there are a couple things you can notice that could make this a little easier though ig

in this specific case

like the axis of symmetry and y-intercept immediately tell you two of those values

[sinful arrow]

so this has 1 double root

is a pst

[patent oak]

Can anyone help me

With 2 problems

It’ll be in #help-8

[cloud stump]

No

[lapis snow]

In chapter 14 of Stewart's Galois Theory 2nd edition (1989), he makes reference to a "well-known trigonometric solution to the cubic". What is this?

[grave marsh]

ig it using the identity
cos(3theta)=4cos³(theta)-3cos(theta)

[winged sand]

can anyone help me with a few trig problems it will be in #help-4

[pine needle]

Is there anyone who can help me with Euclidean Geometry??

[humble pulsar]

dont ask to ask, just post the question

[pine needle]

1.4 i nedd help with

[upper karma]

can someone just confirm this

[pure cape]

thats wrong

[upper karma]

what should it be

[rich wolf]

Horizontal dilation by a factor of pi/8

Not 8/pi

[somber coyote]

dackid (jump king +)

[fast pulsar]

Or is the pythagorean theorem a sufficient condition to form a triangle?

[dark sparrow]

wym

are you asking whether it's possible to have a triplet of [presumably positive] numbers (a,b,c) satisfy a^2 + b^2 = c^2 without there existing a right triangle with sides a, b and c?

@fast pulsar please clarify or face malicious compliance

[fast pulsar]

Correct

[livid moss]

You can just construct it for positive a,b,c by taking the smallest two (say a and b), and drawing a line of length a and then a line of length b perpendicular to the other line.

[fast pulsar]

Ah yes. So the answer is no.

Thx

[livid moss]

Np

[tribal sierra]

Can anyone help me with this problem

[last falcon]

so do you see a right handed triangle

[tiny yacht]

https://math.stackexchange.com/q/4189329/845875

[pure bronze]

When there’s a will, there’s a way 😄

[silk patio]

It’s about reconstruction. A lot of those you can work out in a second by thinking visually

[grave marsh]

tbh all you need is sin(a±b), cos(a±b)

sin is a good boy. so it gives room to cos thus sin(a±b)=sina cosb±sinb cos a.

whereas, cos is kinda devilish, so it keeps most of the space for itself. ie. cos(a±b)=cosa cos b -+ sina sinb

note that cos is devilish so it changes sign

if that helps

[wise pawn]

I never memorized them and failed pre calculus and had to take stats in highschool the next year

I only now know how to produce them because in calculus after learning e^{ix}=cos(x)+i sin(x) I found out we could just derive the sum and difference angle formulas by using exponent rules on e^{i(x+y)} = e^{ix}e^{iy} and then look at the corresponding real and imaginary parts to get the identities

once you know that I can just think "ok the real part will come from cos(x)cos(y) and isin(x)isin(y) which makes i^2=-1"

[tepid crow]

Yeah the original proof with the diagram was a bit confusing

Khan academy has a video on that I think

https://youtu.be/R0EQg9vgbQw

[wise pawn]

here's a funny trig identity lol sin(x-y)sin(x+y)=(sinx-siny)(sinx+siny)

[storm heath]

Does a translation apply to the whole graph or just to the given point/points.

[wise pawn]

your question is a bit vague

if you translate a graph, a graph is just made up of points, so you're translating points

[storm heath]

Yeah

So see

A point on a graph is translated by some amount.

So will another point also on the same graph get translated?

[wise pawn]

depends on how you're translating the points

[storm heath]

Okay.

[surreal bolt]

Translations often mean shifting the entire set of points along a vector. No rotations or weird deformations. If you mean translations in this sense all points get shifted the same distance.

[pure cape]

theres a channel called #chill for that

[silk patio]

One reason cosine has a negative sign is it can be derived from considering the length of a chord in a circle. And to calculate that you need the subtraction of two angles, but then it comes out as plus. So the addition of two angles needs to come out as a minus

[snow flare]

given a 3rd degree polynomial $f(t)=at^3+bt^2+ct+d$, what does it mean geometrically speaking to have a value such as $f(t)<{a,b,c,d}$?

[somber coyote]

bµg

[dark sparrow]

?

what's that supposed to mean?

[snow flare]

i have a curve of y=f(t), if i pick a point p in this space, if its y value is smaller than all the coefficient of the polynomial, what does it mean about p geometrically speaking?

like, is it "below" the roots or something?

am i formulating the problem correctly?

i also have the same question but this time with $min(a,d) <= f(t) <= max(a,d)$

[somber coyote]

bµg

[snow flare]

i'm having a hard time making a relationship between the coefficient values and what i observe geometrically

[acoustic jungle]

why doesn't it just mean nothing

if f(x) = 1*x, what is special about the points with f(x)<1

I don't think there is anything special about them

[snow flare]

it means the point is located in the bottom-left

[acoustic jungle]

ok f(x) = 0

[snow flare]

this is a specific point, i'm looking at ranges

what it means geometrically for points under certain restriction relative to the curve

does it mean the point is below, above, does it have a relationship with the roots, local minimum & maximum?

that sort of stuff

right now i'm observing geometric assumptions made out these constraints, so i'm trying to understand what they mean

iirc the signs of the coefficients can typically be used in descartes rules of sign and has significant implications in the shape of the curve

but here the range checks make use of the value directly, not just the sign

[acoustic jungle]

the rule of signs only give restrictions on the roots

[snow flare]

i guess i'll just try

[acoustic jungle]

why do you think there's a geometric meaning of f(x)<a,b,c,d

what if f(x) = -x

now it's in the top corner

[snow flare]

because i'm observing someone making geometric assumptions out of it

and i'm trying to understand why

and what are these assumptions

[acoustic jungle]

what are the assumptions they made

[snow flare]

that's what i'm trying to understand, but they somehow imply the crossing of the curve

[acoustic jungle]

I have no idea then.

where's their proof

[dark sparrow]

@snow flare where did this sort of condition arise from anyway?

[snow flare]

random code i'm looking at

it basically looks like an optimization for simple cases because it works without

[dark sparrow]

huh?

what problem is the code solving?

[snow flare]

so, the exact assumption is: given $B(t)=at^3+bt^2+ct+d$, if $p_{x}<min(a_{x},b_{x},c_{x},d_{x})$ then $min(a_y,d_y) \leq p_{y} \leq max(a_y,d_y)$ implies 1 intersection if true and 0 if false

and i'm assuming it's an intersection with an horizontal ray

passing by point p

[somber coyote]

bµg

[snow flare]

i'm trying to understand how that works

[fervent valley]

how do u graph this again>

Do u need a gdc for this?

for the graoh

I dont think it will be accurate by hand

[tepid crow]

Pretty sure the general shape is enough with a few precise points like the intercepts and vertex

[rare sierra]

teach me number

Can anyone tell me how to factor correctly?

[dark sparrow]

factor what?

natural numbers? quadratics? polynomials in general?

[rare sierra]

polynomials

[humble pulsar]

ok learn poly division and remainder/factor theorem.

[rare sierra]

Alright thanks

[sly iron]

Can anyone tutor me?

[errant musk]

what do you need?

[haughty cedar]

please help

[upper karma]

done anything?

[upper karma]

Is there an easy way to show a regular dodecahedron exists?

[dark sparrow]

...construct one?

what's the context for this

[upper karma]

constructing one seems really tedious

[dark sparrow]

start with a regular pentagon or something

why do you need to prove the existence of these, exactly?

[upper karma]

my book just seems to assume it exists without proof

i dont feel comfortable accepting something without proof

[dark sparrow]

can you imagine sticking twelve regular pentagons together with three meeting at each vertex?

if you've played ttrpgs, imagine the shape of a d12

[upper karma]

imagination is not a proof

and i cant really imagine it

[dark sparrow]

if you're so insistent, wikipedia gives an explicit vertex list for a suitably scaled regular dodecahedron

or is that insufficient for your majesty?

[upper karma]

seems like hell to verify its a reg dodecahedron. guess theres no easy way

[dark sparrow]

would you like your court jester to start into a song and dance about how the coordinate system exists?

why not question the existence of sqrt(5)?

or 1/2?

or 1?

or negative numbers?

or zero?

[upper karma]

are you okay

[dark sparrow]

or numbers at all?

yknow

[livid moss]

I can send you a picture of a dodecahedron dice I have if that counts as construction

[upper karma]

can construct those easily from set theory and dedekind cuts for example

[livid moss]

What book are you working from?

Like, most books aren't going to formally prove the existence of every object it works with

[upper karma]

Artin, Algebra

[livid moss]

Yeah, an algebra book wouldn't really dedicate the number of pages required to show all geometric objects exist

[dark sparrow]

do you also want to formally prove the existence of the dodecahedron's edge graph

take a graph of 20 vertices named A through T with the following edges:
AB BC CD DE EA AF BG CH DI EJ
FK GK GL HL HM IM IN JN JO FO
KP LQ MR NS OT PQ QR RS ST TP

you can use this edge list to construct a regular dodeca face by face, keeping in mind the edge lengths all have to be 1 and the face diagonals all have to be the golden ratio

ABCDE is one of the twelve faces of the dodeca under this labeling

[upper karma]

Why are you angry? my question was just if there was a simple way to show it exists. constructing one is a tedious way

[upper karma]

<@&268886789983436800>

[rain tangle]

for tan30 would it be necessary to rationalize

or would 1/sqrt(3) also count as a corrct answer?

[tepid crow]

Most places write it as 1/sqrt(3)

[rain tangle]

ok

thanks

[humble pulsar]

No

the need to rationalize is up to the teacher/professor/individual

[somber coyote]

bµg

bµg

[haughty cedar]

no i havent it's pretty confusing and hard for me to understand how to plot the points and follow the steps, are you able to help me out?

[sly iron]

someone to tutor me geometry

10th grade math in the usa

[haughty cedar]

can somebody help me do it.. i wouldn't ask for anything else other than get this one problem i really need right🥺🥺🥺

[wintry tundra]

Idk man we can't really completely tutor you

I could dm you occasionally

I'm also going into 10th but I've taken geometry before so

[haughty cedar]

do you think you can help me on that problem?

[trim breach]

You might want to start by defining points ABC.

Like, with a coordinate system.

Or does it give you specific points for A, B, and C?

[haughty cedar]

A(2, 1), B(3, 3), and C(1, 6);

yes it does i believe that is the points for abc

i don't know where to plot the points

[vocal coral]

no it isn't

[haughty cedar]

to me it is, probably not to you because you are so good at math.

[halcyon mango]

if you're feeling very stuck, maybe go back and review the explanatory chapters in your book and do easier practice problems

it's very normal to be stuck in maths, in fact i'd say something is wrong if you're never stuck

professional mathematicians just know how to be efficiently/constructively stuck, that's the difference

[haughty cedar]

i got it thank you!! sorry i didn't update you guys and really :0 math is just a hard subject for me always been every other subject i'm good

[halcyon mango]

you're trying and you're seeking help which is already great

don't buy into "you're not good", you just happen to be in a little deep that's all

[haughty cedar]

a little deep?

[halcyon mango]

like, deep past the point of your own technique in whichever topics

but maybe it's only a few review sessions away from being not so bad for you

[haughty cedar]

yeah and thanks for being so nice and being here for me!

youre so nice 🥺🥺😭💙

[halcyon mango]

incidentally, people often have interesting relations with topics they struggle with at first

stephen wolfram is famously self-admittedly bad at calculations

but it led him to do a lot of work in computational research

[halcyon mango]

no worries, good luck!

[haughty cedar]

wowwww and thank you!! got this 🙂

[coarse hare]

do you guys like triagles

[dusty flicker]

Yes

[coarse hare]

hahaha

[dusty flicker]

😦

[vocal coral]

drinks @dusty flicker

[small trail]

Can someone tell me how this geometrical figure is called? (I sketched it in Paint).

It's basically a cuboid, but the backside can be either smaller or bigger than the frontside, resulting in angles other than 90 degrees.
Would it maybe be called 3D trapezoid or something like that?

[pure cape]

trapezoidal prism?

[rich wolf]

@small trail sketch the side view

That should make it very clear

[pure cape]

not sure if trapezoidal is the right adjective

[rich wolf]

Sketch the side view

[dark sparrow]

is it like a pyramid with the top chopped off?

[small trail]

I'll try to sketch the side view

To make it easier to understand. Think about two random rectangles in a 3D space and connect all 4 edges of both with each other. What you then get is what I am looking for.

[dark sparrow]

two random rectangles... so not necessarily similar?

[small trail]

exactly

[rich wolf]

What if the planes they lie on are perpendicular

[small trail]

Ok I have to give more info. The rectangles are parallel.

(Left and right on my second picture)

[rich wolf]

Trapezoidal prism

Look at top and bottom

And the rectangles would be similar

[small trail]

Like this?

With top and bottom on this picture being my two rectangles

[pure cape]

tho actually, its not a trapezoidal prism since prism is named based on their base?

[rich wolf]

Base can be whatever side you choose

[pure cape]

hmm

[rich wolf]

It's the same shape regardless of rotation

[pure cape]

oh wait, it says prism are named based on the cross section

i guess its a trapezoidal prism then

[small trail]

Thanks for the help

[halcyon mango]

sounds like the angles are the same (prob accounting for a factor of +2πn?)

[snow flare]

Can anyone help me figuring out the T here?

[somber coyote]

bµg

bµg

[snow flare]

any hint?

[fossil yacht]

just look at where the basis vectors naturally go to

u maps to (1,0) and v maps to (0,1).
the linear transformation will be completely determined by those two values

[somber coyote]

bµg

[snow flare]

i'm sorry it feels like it's something really retarded but i struggle quite a bit with it

(do i need to ask in #linear-algebra instead?)

it feels like a geometric problem to me (which could be solved with linear algebra)

[fossil yacht]

the linear transformation you’re looking for is the unique linear transformation T which is determined by T(u) = (1,0) and T(v) = (0,1). that’s all the information required

[snow flare]

@fossil yacht oh i can indeed make a simple system with that, let me try; thanks

[fossil yacht]

to get P’ you just write P in terms of u and v, so au + bv for some scalars a and b, then look at what happens to T(au + bv)

[snow flare]

@fossil yacht i followed your initial direction; am i doing something retarded or does that look somehow ok?

(sorry for the wild layout)

[fossil yacht]

i’m not really sure what u are trying to solve here.

do u mind explaining

[snow flare]

trying to get T according to u and v

so that i can use the transform

i tried to make a system knowing T(u)=(1,0) and T(v)=(0,1)

i assumed it could take the form of a basic polynomial

(that's the form i'm actually looking for)

and then i "solved" the system

[snow flare]

https://www.wolframalpha.com/input/?i=solve+{p_0%3Dc_0%2Bx*u_0%2By*v_0%2C+p_1%3Dc_1%2Bx*u_1%2By*v_1}+for+{x%2Cy}

this might actually be a more appropriate solution

the values are a bit wild though

[weary drift]

draw an appropriate right triangle

[glad falcon]

@fossil yacht

How do

[fossil yacht]

so you need to use similar triangles

do you agree that cf/2 is the area of triangle ABC?

[glad falcon]

No

How is cf/2 the area

[fossil yacht]

base times height over 2

[glad falcon]

But height is AC no?

And isn’t base a

[fossil yacht]

there are multiple bases and heights

any side can be considered a base

[glad falcon]

You know it’s case sensitive so did you mean the correct C

Yes. I agree with you

[fossil yacht]

i meant that the base is c and the height is f

[glad falcon]

Area of ABC is cf/2

I agree, now…

[fossil yacht]

okay cool

so now we need to get some information on some other sides

do you see any ways to do this using similar triangles?

[glad falcon]

f is 6 c is 10

[fossil yacht]

nice. we can also say that ab/2 = cf/2 = 30

[glad falcon]

Oh yes

[fossil yacht]

AC is proportional to BC and AB

[glad falcon]

Then…

I got you

[fossil yacht]

but you can use ratios of side lengths to get some more information

for example, b/d = a/f = c/b

[glad falcon]

Yes

[fossil yacht]

try playing around with that for a minute. also don’t forget about the pythagorean theorem

[glad falcon]

b/d = a/6 = 10/b

ab = 60

I got no clue sorry

[glad falcon]

^

[fossil yacht]

ok so we should have d/f = f/e, right?

[glad falcon]

Yes

[fossil yacht]

so we get de = f^2 = 36

[glad falcon]

Yes

[fossil yacht]

then you can substitute this in for d + e = 10

[glad falcon]

How?

[fossil yacht]

if d + e = 10 then ed + e^2 = 10e

and you know ed

so now it’s just a quadratic equation

u can solve for e i think

[glad falcon]

Ed is 36

How you get 10e

[fossil yacht]

you have e^2 - 10e + 36 = 0

so use the quadratic formula

[glad falcon]

How you get 100e from

Yes

[fossil yacht]

yea so e + d = 10

and you just multiply both sides by e

[glad falcon]

But the Ed is 10e

[fossil yacht]

how is ed equal to 10e?

[glad falcon]

e(e + d) = 10e

[fossil yacht]

yes. now e^2 + ed = 10e

[glad falcon]

Yes

[fossil yacht]

and ed=36

[glad falcon]

Then solve for E

[fossil yacht]

right

[glad falcon]

Which is

[fossil yacht]

but this is odd because there are no real roots…

so… either we messed up or the problem is jank

let’s back track.
step one: d/f = f/e

[glad falcon]

Yes

[fossil yacht]

this works because of similar triangles

[glad falcon]

Yes

Then

[fossil yacht]

okay.
step 2: ed = f^2 = 36

[glad falcon]

Yes

[fossil yacht]

just algebra

step three: e + d = 10 implies that e^2 + ed = 10e

[glad falcon]

Yes

[fossil yacht]

so e^2 - 10e + 36 = 0…

[glad falcon]

Yes

[fossil yacht]

and e is a complex number so we broke geometry

[glad falcon]

So now?

[fossil yacht]

dude i got no clue. the problem has to be wrong

[glad falcon]

It’s not lol

[fossil yacht]

lol we didn’t make any errors we just checked

is there an answer key?

[wintry tundra]

How is e complex?

[fossil yacht]

just plot the graph. it’s always positive. no real roots

[wintry tundra]

So e^2 = -36 + 10e

Oh ok

If its always positive

Then there are always real roots

Or so I've been told

[fossil yacht]

no if the graph is always positive or always negative, then there are NOT any real roots.

[wintry tundra]

Sometimes I wish I had a brain

Ima leave u alone

[fossil yacht]

lol it’s cool man haha

[glad falcon]

Wait the question is find the area of the triangle

[wintry tundra]

I was thinking roots as in square roots

[thorny terrace]

lol no roots as in zeroes/solutions

[fossil yacht]

this whole question doesn’t make sense

there isn’t even a real solution to ab = 60 and a^2 + b^2 = 100

this question is doomed lmao

like the triangle in the picture just doesn’t exist. are you sure f is 6 and c is 10?

[fallen osprey]

I had to fill in the spots where i circled, I'm not sure if I did them right, can anyone help 😦

<@&286206848099549185>

[fallow minnow]

a b c volume doesent worrk on mu x[x-y=0.9y

[crystal quail]

Seems right to me

[fallen osprey]

help

[thorny terrace]

no

yes lol what's up

[idle meteor]

whats a cool use for trigonometry in programming? (Ik this isn't a programming discord server but i'm going to still ask)

[crimson sluice]

which language are you familiar with

language(s)

[idle meteor]

python, javascript

[idle meteor]

idk if i did something wrong (sorry for bad handwriting) but for me it isn't a complex number idk

i probably did something wrong 100%

[idle meteor]

e = 10-(5 plus or minus sqrt(19))

for me

[old fable]

Why does it matter that cosy > 0

For the derivative of arcsin

[dark sparrow]

so that you know cos(y) = +sqrt(1 - sin^2(y)) and not ±sqrt(1 - sin^2(y))

[old fable]

Ohhhh ok

Thanks

[halcyon mango]

yea with the inverse functions the domain/range issue is important to get a hang of, it is worth thinking about

[old fable]

What has the fact that y doesn’t equal π/2 to do with the abs(x) in the denominator

[dusty fulcrum]

I have done this one, but I just wanna double check that im acc right lol. You can take a crack at it if u want, but u dont need to write an A4 page lmao. Just an inequality will do.

[halcyon mango]

when you're doing implicit differentiation to get the derivative, you have to be mindful of the range of arcsecx, particularly when you take square roots. This is maybe a similar idea of the formulas for half-angles depending on the quadrant (so you cannot take both + and -)

[old fable]

Yeah ik that

But for this derivative you need to keep in mind the domain of sec y

[dusty fulcrum]

this is what i got btw

[halcyon mango]

what's cos(π/2)?

[old fable]

And I’m not seeing what the y doesn’t equal π/2 has to do with it

[dusty fulcrum]

||-0.25 < k <= 0||

[old fable]

0

[halcyon mango]

there you go

[old fable]

But why is it abs(x)

Because sec y > 0

?

[halcyon mango]

when you do the identity tany = +- sqrt(sec^2(y)-1) it has to be in the correct quadrant, absolute value guarantees this

there's also another derivation with logs i think

i can't think of it off the top of my head

[old fable]

Ohhh ok

[halcyon mango]

no i'm making it up about logs

lol

[old fable]

And what does the fact that y doesn’t equal π/2 have to do with that?

[halcyon mango]

sec y = 1/cos y. so π/2 would break it

[old fable]

Ok so with abs(x) it doesn’t ?

[halcyon mango]

well the image you linked is talking about the definition of arcsecant

and one way to define it is to limit it accordingly

y = arcsec <--> sec y = x, 0 <= y <= π... but you also need to stipulate y≠π/2

if you graph it it becomes really evident

try graphing all the inverse functions without desmos first

[vernal quartz]

Please don't solve any of the questions for me

I just don't understand what the first statement means

I assumed it would be what values of θ give r = 0 but it appears not to be so

[light charm]

I think the first statement is meant to be a general case when finding specific values can be harder than the question

Maybe you need to find properties of sin and cos of values that can create double angle formula etc

[vernal quartz]

I've just rewritten it in terms of tan1/2θ

I have a 4th order polynomial so perhaps I'm on the right track

[fossil yacht]

yea i can’t read that, but idk. something about that problem feels off

[crystal quail]

My Geometry book states the following congruency test for right triangles:
"if the hypotenuse and an acute angle of one triangle are congruent to respectively the hypotenuse and an acute angle of the other, [the two triangles are congruent]"

Why the need for the word "respectively" in this case. In most cases thusfar in the book it makes sense, but in a right triangle it seems to not make a difference.

Say in one triangle the congruent acute angle is between the hypotenuse and the vertical leg, while in the other triangle the congruent acute angle is between the hypotenuse and the base. They would simply form two triangles that are mirror images of each other, therefore would be congruent. So if it doesn't matter which side has the congruent angle why would we say "respectively" here?

[dark sparrow]

it's a style thing probably

"if X and Y are congruent to X' and Y' respectively" even if X and Y are different types of objects

[crystal quail]

So would it still be accurate to say the same statement but without "respectively"? I'm writing a text for students and I want to be very specific with my wording and explanations

[dark sparrow]

i think it would sound somewhat weird from a style pov

[crystal quail]

Oh wait this makes sense. You want to clarify that the hypotenuse of one triangle is not congruent to the...well angle. Like you said even if other objects. that 100% makes sense now

tyty

[vagrant barn]

i am very confused im not so sure if i've got the right answer here

[surreal bolt]

Er what’s lateral area?

[versed river]

nope

[surreal bolt]

What is your formula / definition?

[versed river]

its the area not including the bases/tops of an object

which is a weird definition but 🤷‍♂️ some places use it

in this case the base is considered to be the octagon

[surreal bolt]

It has its uses.

Yeah the picture is misleading.

You are trying to find the areas of all of the triangles not the base.

[vagrant barn]

ooh ok

[surreal bolt]

The base is gray to accentuate the 3D perspective, but isn’t the region you are asked about.

[vagrant barn]

ok yeah that makes sense

[bitter iron]

Please help and show working out

[surreal bolt]

Do you like rectangles? (Hint)

[bitter iron]

Yes

[surreal bolt]

Hint … draw a rectangle in the picture provided.

[bitter iron]

Okay

[surreal bolt]

Label all lengths.

5.5 is split into what and what else? What is the length of the red line segment?

[bitter iron]

Red line segment is 4.2

[surreal bolt]

What about the 5.5? It is split.

[bitter iron]

2

[surreal bolt]

Okay Pythagorean theorem?

[bitter iron]

4.7

[surreal bolt]

Yeah. That is what I got. Hope it is right.

[drowsy sparrow]

please

can anyone help me solve this problem?

[spiral hawk]

@drowsy sparrow So basically, being vertexs in quadrants 3 & 4 make the triangle inside of the eclipse to look like this, then once you have your triangle you can split it into to right angle triangles, and solve for area of the triangle

[wintry tundra]

What do u need help with

[worn sparrow]

im ac done with that

but im stuck on this

[wintry tundra]

Ok that took 20 minutes

Uhh I got $\tan^2{\theta} + 2 = \sec^2{\theta}$

[somber coyote]

visual of Petter's ascendance

[wintry tundra]

Maybe bc I'm bad at math but idk

[worn sparrow]

What’d u simplify

[wintry tundra]

All of it

[worn sparrow]

Which tho

I know on left side

U can simplify to just sin^2

And first part on right side is sin^2 / cos^4

[wintry tundra]

Well I know that sin^2/cos^4 is just sin^2/cos^2 * 1/cos^2

Which I can turn into tan^2 sec^2

[worn sparrow]

wat about the other part on the right side

[upper karma]

@shrewd delta

can you help me sir

[shrewd delta]

Why tag me?

[upper karma]

idk

just saw someone tagging

so i did the same

[shrewd delta]

Well don't tag people unless they say that you can you can wait a while, someone will help with the problem eventually (if no one does after a while, you can tag helpers)

See #❓how-to-get-help

[upper karma]

sprry

mold

[shrewd delta]

np

[upper karma]

@upper karma do you still need help?

[fossil yacht]

this is just an application of the law of cosines.
$$c^2 = a^2 + b^2 - 2ab\cos(C)$$ where $c$ is the unknown side length, $a=95.5$ and $b=83.1$ are the known side lengths, $C = 101^{\circ}$ is the angle opposite of side $c$

[somber coyote]

coycoy

[edgy juniper]

is there a rule that
c b a

•   =       -     =      -
  

sinC sinB sinA

if the bottom lime was “a”

[dark sparrow]

you really should not be writing fractions like this

but yes, the rule you wrote exists and even has a name: the law of sines

[edgy juniper]

was just in a rush and didnt know how to write on my phone

[dark sparrow]

c/sin(C) = b/sin(B) = a/sin(A)

[subtle palm]

yea this looks so confusing

like they are referring to the same triangle