geometry-and-trigonometry

[cedar prawn]

but I like being able to have visual metaphors

Anyway, I would like to learn enough math to figure out how to discover whatever properties my hypothetical plane composed of orthogonal helices may have

or if it would even be any different than a normal old Euclidean plane

[vapid kettle]

well

you'd have to define different

[cedar prawn]

I feel like it would be different but it's just two lines multiplied together, it just so happens that they are corkscrews

[vapid kettle]

since topologically

it isn't :p

[cedar prawn]

Yeah topologically it is just a plane

But geometrically it would be very different

I'm pretty sure that in the form I envision it, it cannot be embedded in threespace

whereas a normal plane surely can

but it could still be sort of unraveled into a flat plane if you are able to use topology on it

so this is definitely a geometrical thing I am interested in

not a topological one

I just have to use topological terms to think about it, so far, because they're easier

[vapid kettle]

well the only structure I can think of

of any use that's more geometric

would be that of diffeomorphism

and the way you're talking

sounds like that's the structure you're looking for

[cedar prawn]

I shall look it up... all these words head spins

[vapid kettle]

and I don't know much in the way of differential geometry

but uh

I think your thing is diffeomorphic to the plane as well

but yeah

your intuition for punctured plane = cylinder

mostly works

towards a real proof

but problem with only working with intuitions is

you can get into arguments which seem to work but don't

or infamous false proofs

[cedar prawn]

Well, a failed intuition can teach a lot too

because when you realize why you're wrong... poof! You learn a mistake never to make again

[vapid kettle]

visualization is very useful though ~

and it's not limited to geometry/topology

working with SO_3 today on some sort of algebraic stuff

/differential stuff

and was trying to visualize it using

projective plane with S1 bundle attached everywhere :p

[cedar prawn]

But I know that if you take out the origin, the length from the next point outside it to infinity is equal in size to the real line as a whole - it has the same cardinality - so you could make a one to one mapping from radii around the origin, to locations on a real line; and it is already obvious you could make a one to one mapping from an angle on a circle, to a different angle, so I probably if I knew the right terms to use could write a proof of that cylinder thing

[vapid kettle]

probably ~

[cedar prawn]

A cylinder is just a collection of 1 real line for each angle around a circle

and all the distances above 0 from the origin to an infinite distance away are equal in length to a real line

[vapid kettle]

but there are times where there seems to be an easy intuitive explanation

[cedar prawn]

so it should work

[vapid kettle]

but it doesn't translate well to proofs

or close

[cedar prawn]

Of course. That's why it's important to doubt oneself 😃

and always look for ways to disprove your own idea

[vapid kettle]

ok ;p

[cedar prawn]

By the way this all started when I was thinking of how spheres are finite planes with a set positive curvature everywhere, and wondered "can I 'unroll' them to be infinite but still have the same radius?"

Thinking in terms of two dimensions I noticed that the infinitely long analogue of a circle is a spiral

so I started looking into logarithmic spirals, but their radius is perpetually changing so I thought that wouldn't be helpful - something which is a spiral in both dimensions wouldn't have a set curvature

then I thought about helices

and realized they might do it

but I still have no idea how to prove it of course, or figure out the ramifications XD

[vapid kettle]

mmm

the questions and reasoning here are a bit too vacuous for me

but I encourage you to have fun exploring ideas ~

as one should

[cedar prawn]

"vacuous"?

[haughty prawn]

Oops, I see that I misrememebered, I meant C^2-(0,0)

[cedar prawn]

How is it "vacuous" to think abstractly in the absence of all the serious mathy knowledge I haven't developed yet?

[vapid kettle]

ah googled the word

meant vague/ill defined

[cedar prawn]

Oh okay

[vapid kettle]

was using the word wrong ~

[cedar prawn]

:3

[vapid kettle]

sorry bout that

definitely not what I meant xD

[haughty prawn]

Vague?

Oops you said that already...

My mind is not thinking coherently rn

[vapid kettle]

~

sim

[cedar prawn]

btw it's driving me crazy, the pronunciation there should be /'væ.kju.əs/. I hate this damned English phonetics system used in dictionaries online... use IPA people - it's not that hard!

[vapid kettle]

other thing

I'd suggest adding "bounded" to your vocabulary ~

since it seems to be what you mean when you say "finite"

just a useful way to communicate

[honest socket]

hey peeps, in some piece of code I met this formula

does anyone know what does it produce?

[dark sparrow]

cosine of the angle between sides A and C

[honest socket]

😑 too fast, how is it called?

[dark sparrow]

it's a rephrased version of the law of cosines

[honest socket]

okay, thanks

[honest socket]

i got it, ty

what will happen in this case?

[dark sparrow]

everything breaks and the above formula makes no sense

[honest socket]

B thing points at point P

ohhhh, intradasting

actually i wrote a message, but it wasn't send...

so, what did i do wrong?

about the formula

[dark sparrow]

you didn't

[azure storm]

(except you shoulda have wrote cos C = ... at the last line)

[dark sparrow]

=tex \frac{c^2 - a^2 - b^2}{-2ab} = \frac{a^2 + b^2 - c^2}{2ab}

[honest socket]

oooh i c, it must be multiplied by

=tex \frac{-1}{-1}

[azure storm]

yes thats a way to prove it

or just :

=tex \frac{1}{-1}=-1

[azure storm]

Which just leads to do :

=tex \frac{a}{-b}=\frac{-a}{b}=-\frac{a}{b}

[honest socket]

uh, i'm being stupid

what does the second formula do?

[dark sparrow]

it's equal to -cos(a,b)

-1 times the cosine of the angle between a and b, that is

[honest socket]

wait, i can't use evolved 🤔 smile, that's disappointment
also, it feels like i don't understand something, because i see code of calculating cos and -cos, and instead of multiplying the cos by -1 it calculates -cos with that formula

but let's check the values, maybe there is something wrong in my understanding of situation

[dark sparrow]

you do realize that cos(θ) * (-1) and -cos(θ) are literally the same thing

right?

[honest socket]

yep

that's what i'm saying, cos(theta) * -1 is cheaper than calculating it by the formula

okay, i just misunderstood the code, because when cos is 0.81 the second cos is 0.34

hm... that's interesting, i just pluged all the date in calc and it didn't return -cos when i used the second formula, though it returned the value my code returns

here is data

^ result for the second formula

result for the first formula

=tex 0.34!=0.81

[dark sparrow]

=tex 0.34 \neq 0.81

😛

[honest socket]

i thought about not xD

[dark sparrow]

not equal

[honest socket]

i c i c

[dark sparrow]

=tex \Gamma

[dark sparrow]

😛

[azure storm]

letter = small, Letter=big, varletter=good looking

=tex \sigma \Sigma \varsigma

[dark sparrow]

\varsigma is the sigma greeks use at the end of words

lol

[azure storm]

idc it'll be good-looking-and-usable-for-physics to me!

=tex \phi \Phi \varphi

[dark sparrow]

yes @sour briar

[azure storm]

only the 3rd one looks 💯

[dark sparrow]

this only happens with sigma

@azure storm true af

[charred spear]

Support the bot on Patreon: https://www.patreon.com/dxsmiley

[azure storm]

in what case do you use which s?

how the heck do you draw that without it looking like a typical s?

ah

[azure storm]

my eyes hurt

[dark sparrow]

christ

[honest socket]

okay, i guess the second vlaue is cosine of angle between a and b

[azure storm]

"Introduction Warning : Before Reading this book you should have read analysis intro. After reading it, you should go to the eye doctor"

[cedar prawn]

goodness that font is nigh unreadable

personally I prefer Consolas, best font ever

I love fonts where all the letters are the same size

but if I have to have a "normal" looking one I choose Calibri or Sylfaen

I guess computer fonts didn't exist back then though XD

[cedar prawn]

One of my many weird little goals is to someday create a "perfect font", and yes I know, perfect is different for everyone... but to me that would be a monospace font where each character is as different as possible from all the others, and composed of nothing but lines and circular arcs

[dire bronze]

hey i know 90o counterclockwise rotation is (x,y) to (-y,x) and my question is that should i write my point like (-y,x) (expect with numbers) because my friend is saying yes but i dont believe him since usussaly x goes first

[surreal bolt]

the x-coordinate goes first (left for you crazy non-mandarin writers) ... 😃 BUT the x value could be anything. If I tell you x = z and y = a, hopefully you wouldn't tell me that the coordinate is (a, z) just because z is after a. Similarly, if I told you y = q, would the coordinate be (x, q) or (q, x)?

So in this case we have (-y, x) which means the new x-coordinate is the additive opposite of the original y-coordinate and the new y-coordinate was the old x-coordinate.

[dire bronze]

oh that makes more sense

thank you

[surreal bolt]

Quite welcome.

[dire bronze]

wait

why did it send here

wut

[surreal bolt]

Yeah it is A., but I am sure you want to know why 😃

[dire bronze]

yeah

[surreal bolt]

So the easiest way is to just take the paper you have and rotate it and see if you can make it match the 4 choices

[dire bronze]

ok

[surreal bolt]

um is that sufficient 😃

[golden geyser]

I can not for the life of figure this question out, help please 😄

[surreal bolt]

hmm can you draw the figure first 😃

(it helps a lot when starting out with these ... eventually you may be able to solve them in your head)

[brazen roost]

Translate to math first

My english is shit

[surreal bolt]

The answer is .... 36 😃

kidding know what you're doing.

[brazen roost]

No, the answer is 42

What's the definition of opposite rays again?

[golden geyser]

rays that start at the same point and go in different directions

I think

[brazen roost]

I think

45 degrees

[surreal bolt]

Look it up 😃 not a hard question for the internet 😃

[brazen roost]

Browser is RAM intensive

Don't wanna

[surreal bolt]

um .... okay

Just to make absolutely sure ... yes opposite rays point directly away from a common point.

[brazen roost]

So

180 degrees?

[surreal bolt]

I believe so.

[brazen roost]

Then 180 / 4 => 45

[surreal bolt]

Rascal ... Draw The Picture!

[brazen roost]

But I don't wanna

And I'm doing a biology project

My hands are less than clean atm

[surreal bolt]

lol if you don't need to know geometry or feel you don't ... or just don't have time to do dead cat surgery and have time later to do geometry ... well I cannot help ya. 😃

[brazen roost]

It's not me

tdfirelord needs geometry

[golden geyser]

lol

its me

and I think I kinda figured it out

[brazen roost]

Mario

[golden geyser]

its 36 right?

[brazen roost]

...

[surreal bolt]

oh bah.

😦

[brazen roost]

It's 45

In case you haven't figured it out yet

[surreal bolt]

ya sure?

[brazen roost]

Juuust in case

[surreal bolt]

eesh 😃 have good study habits.

[brazen roost]

What does the m in front of the angle mean?

[golden geyser]

measure?

[brazen roost]

Ok

My answer is still 45

[surreal bolt]

lol care to explain it?

[brazen roost]

Since WX and WZ are opposite rays, m<XWZ = 180

[surreal bolt]

Agreed 😃

[golden geyser]

yup thats what I got too

[brazen roost]

And since m<XWZ = 4 m<YWZ, Then m<YWZ = m<XWZ / 4

Which is equal to 180 / 4

Which is 45

QED

[surreal bolt]

eh you either copied the problem wrong or I can't read.

[brazen roost]

You can't read

[golden geyser]

I cant copy it wrong its a picture from a textbook 😄

[surreal bolt]

XWZ is not mentioned in the equation 😃

[brazen roost]

It is

Oh wait

I can't read, it seems

[surreal bolt]

Don't make it a habit or a curse

[brazen roost]

XWY

[surreal bolt]

😃 that is my study skills advice

[brazen roost]

Fuck you similar looking alphabet

Or curse you

Either one works

So

180 / 5

36 yeah

[golden geyser]

kk

[brazen roost]

Dammit

I guess I'm too used to calculus

[surreal bolt]

Heh there is a lot of math to know up to Calculus. Can you do old-style high school induction problems?

[brazen roost]

Prove f(1), then prove f(n), then prove f(n+1), Thus f works for any n

[surreal bolt]

😃 oh I suppose that is fine.

[brazen roost]

What grade does america start doing integration?

[surreal bolt]

It is not required for high school graduation in most high schools.

[brazen roost]

Hmm

What's the highest math you do in america?

[surreal bolt]

in high school?

[brazen roost]

Ye

[surreal bolt]

to graduate or just what is the highest math?

[brazen roost]

Both

[surreal bolt]

To graduate, I think most states require Algebra II. But the sky is the limit as far as taking college level classes. I mean I know a 14 year old went to MIT (know of him). Probably finished his Bachelor's by 18. 😃 Does 18 count as still high school?

[brazen roost]

Oh wow

[surreal bolt]

Hehe. Poor guy likely couldn't drive let alone drink.

[brazen roost]

lol

[surreal bolt]

Okay I think that one disparaging statement about a Boston Beaver is enough -- especially one with amazing academic credentials. I'm out. Later!

[brazen roost]

cya

[honest socket]

let's say we have an arbitrary triangle, we know length of 2 sides, also we know the point where those 2 sides meet, how can we calculate length of the 3rd side?

ug, i forgot, the angle between A and unknown side is 90 degrees

and i guess i already now the solution...

[honest socket]

ahem... and how to do the same but when the angle between A and ? is unknown?

i mean when it's arbitrary

so we know the angle, but it's not 90 degrees

[dark sparrow]

b^2 = a^2 + x^2 - 2ax cos(θ)

where x is your unknown side and θ is your angle

[honest socket]

oh i c, it looks like some weird form of law of cosine with X as unknown side

[brazen roost]

You can replace any variable with any other variable

It's not weird in any way

[honest socket]

i c i c

[dark sparrow]

=tex AB^2 = AC^2 + BC^2 - 2 \cdot AC \cdot BC \cdot \cos(ACB)

[rugged moat]

5cos(A) +12sin(A)=13
What is the value of tan(A)

[dark sparrow]

oh boy

okay, so

[rugged moat]

Tan(A)-Tan(2A)=1
Sin^2(2A) + tan^2(2A)=?

[dark sparrow]

err okay let's do these one by one

[rugged moat]

Sure

[dark sparrow]

okay so

=tex \frac{5}{13}\cos(a) + \frac{12}{13}\sin(a) = 1

[dark sparrow]

does this make sense?

[rugged moat]

You divided by 13

[dark sparrow]

yeah

now

(5/13)^2 + (12/13)^2 = 1

so 5/13 and 12/13 are the cosine and sine of some angle

i'll call this angle b

we don't know its value, but we actually don't need it

=tex \cos(b)\cos(a) + \sin(b)\sin(a) = 1

[dark sparrow]

do you follow?

[rugged moat]

Yeah, I follow

[dark sparrow]

cos(b-a) = 1

b - a = 0

so b = a, and thus tan(b) = tan(a)

does that make sense?

[rugged moat]

Wait

Got it

[dark sparrow]

alright

so we know that sin(b) = 12/13 and cos(b) = 5/13, since that's how we defined b

so tan(b) is just 12/5

and that's your answer

[rugged moat]

Hmmmmmm, thanks

Next problem?

Tan(A)-Tan(2A)=1
Sin^2(2A) + tan^2(2A)=?

@dark sparrow

[dark sparrow]

i'm thinking, give me a moment

tan(2a) - tan(a) = -1

okay

sin^2(2a) + tan^2(2a) = sin^2(2a) + sin^2(2a)/cos^2(2a)

= sin^2(2a) * (1 + 1/cos^2(2a))

er

wait

shit

okay this is trickier than i thought

hold on

[rugged moat]

Sure

Sin^2(A)+cos^2(B)=2
Cos((A+B)/2)=?

---

Tan(@)+cot(@)=2
Tan^5(@)+ cot^10(@)=?

---

Sin(17°)=x/y
Sec(17°) - sin(73°)=?

You there?

@dark sparrow

.........

@dark sparrow

Sure

If anyone finds the solution ping me please

[dark sparrow]

ARGH2L23;5R2SRSEDOPS

DER;GSRTHGAEGKOSDJFGHSD\

SORRY

!!!!!!!!

arrgh

anyway

tan(x) + cot(x) = 2 can only happen if tan(x) = cot(x) = 1

since they're reciprocals of each other

a + 1/a ≥ 2 for all positive a

@rugged moat

SORRY I WAS ABSENT SO LONG ASDFNASJDFNASJDKFALDKF

[rare talon]

Anyway tan (2x) - tan(x) = -1 ?

Anyone solved that?

[maiden trench]

I'm working on it but it seems like it has no solutions. Although I might have made a mistake in my algebra so I'm double checking.
Oops, I did.

[rare talon]

I got sin(x) = -cos(x) . cos(2x) so far

[vapid kettle]

someone should graph it

[rare talon]

Oh

I think I get it

tan(2a) - tan(a) = sin(2a)/cos(2a) - sin(a)/cos(a)
= {sin(2a) cos(a) - sin(a) cos(2a)}/(cos(2a)cos(a))
= sin(a) / {cos(2a)cos(a)}

Then the eqution becomes

*equation

Actually wait

I got this so far:
cos^2 (2x) + cos(2x) + sin(2x) = 0

idk if that's right -__-

[maiden trench]

I used my calculator to find a numerical value for x and that equation is correct when I plug the value in.

[rare talon]

Okay hanks

*thanks

I am not in the wrong track then

(I mean, no mistakes so far)

[rugged moat]

I am bavk

Back*

[rare talon]

Is one of the solution , cos(x) = 0 ?

I'm lazy B(

[dark sparrow]

given how tan(x) is involved i'm having huge doubts about that

[rare talon]

o sry then it's a leftover

I used many manipulations

And yeah some of them is multiplying by cos(x)

Then cos(x) won't be 0

I got the other solution like this though:
cos(2x) . cos(x) + sin(x) = 0

[quiet orbit]

what's the problem?

[rugged moat]

Tan(A)-Tan(2A)=1
Sin^2(2A) + tan^2(2A)=?

[quiet orbit]

alright, thanks

[rugged moat]

Anyone made any progress

?

[rare talon]

Well I managed to simplify the tan(a) - tan(2a) stuff

And I feel fishy about that one

Can you take a pic on all of the problems? 😛

[quiet orbit]

i guess it can be reduced to:

t^3 - t^2 + t + 1 = 0

1/t^2 - t^2 = ?

t != +/-1

[rare talon]

RIP I can't read

But well I'm still trying that prob

dw

[rugged moat]

Both of my cameras have moisture in them but my front camera is a lot less affected

I searched the problem on Google but got no hits

[maiden trench]

I plugged it in there... it's got a solution but it's certainly not nice.

[rugged moat]

Can you try tan(A)×tan(2a)=1

[brazen roost]

What's the original problem?

[rugged moat]

Tan(A)-Tan(2A)=1
Sin^2(2A) + tan^2(2A)=?

[rare talon]

I actually think we might not want to find A

[brazen roost]

euler's identity!

[rugged moat]

It's written in a wierd way

What is Euler's identity?

[brazen roost]

e^ix = cos x + i sin x

You can derive all sorts of things from there

[rugged moat]

What is e and i?

Btw it should be a really easy solution

[brazen roost]

i is the unit imaginary number

[rare talon]

@brazen roost how would you do that with euler?

[maiden trench]

I don't think Euler's identity is helpful here.
If it actually is tan(x)tan(2x) then it's a lot simpler.

[rare talon]

Anyway I think he wants a manipulative trig solution

[rugged moat]

I am in grade/year 10 and am just introduced to trigonometry

[brazen roost]

Turn everything into an exponential form then algebra the fuck out of it

[rugged moat]

Maybe you all are thinking too hard

[brazen roost]

I guess there's a unit circle solution to this

[rugged moat]

Now, what's a circle solution?

[brazen roost]

If tan(2A) - tan(A) = 1

Actually

nvm

A unit circle

Is a very helpful diagram

Of trig functions

[quiet orbit]

http://mathb.in/150542

[brazen roost]

And it's like midnight here, so my brain isn't working properly

[quiet orbit]

so you have to find 1/t^2 - t^2 where t is a root of that polynomial, you can write 1/t^2 in terms of powers of t actually

[rugged moat]

It is math processing error everywhere

[brazen roost]

You're welcome

[rugged moat]

Can you please explain to me the first step

[brazen roost]

double angle expansion

With tan

[rare talon]

tan(2x) = tan(x+x) = use your tan formula

[rugged moat]

We haven't learnt that, yet

[brazen roost]

Ok

Cool

Then why the hell does that question appear

[rugged moat]

You all are thinking too hard

It should have an easy solution

[brazen roost]

Tan(A)-Tan(2A)=1
Sin^2(2A) + tan^2(2A)=?
You say

Hold on

One last question before sleep

[rare talon]

I don't, I just use trig manipulation

And it leads nowhere

[rugged moat]

Try tan(A)×tan(2A)

Sin^2(A)+cos^2(B)=2
Cos((A+B)/2)=?

Atleast tell me this one

[rare talon]

Note that sin^2(A) has max value of 1

And also cos^2(b) has max value of 1

[rugged moat]

And this----------
Sin(17°)=x/y
Sec(17°) - sin(73°)=?

[rare talon]

Then sin^2(A) = 1
And cos^2(B) = 1

Solve for a and b

[rugged moat]

I want to simplify it

[rare talon]

What

[rugged moat]

a and b would be 90 and 0

[rare talon]

That's one of the example of the value

[rugged moat]

It would be way too easy

[rare talon]

Yeah indeed it's easy

Lol

Sin^2(A)+cos^2(B)=2 this already indicates
sin^2(A) = 1 and cos^2(B) = 1

It can't be other value other than that

And this----------
Sin(17°)=x/y
Sec(17°) - sin(73°)=?

Hint: Draw a right triangle model

Just use the ratio on that one

[brazen roost]

That would be 0

Easy

[rugged moat]

I drew a triangle

But in the end it came in the form of tan and cos

[rare talon]

Remember sec(17°) = 1/cos(17°)

[brazen roost]

Rule of thumb: if the angle is anything other than 30, 45, 60, 90, and other multiples of those, it probably doesn't matter

[rare talon]

And sin(73°) = cos(17°)

[rugged moat]

I need a numerical value

I remember

[rare talon]

Given the problem on that nature

[rugged moat]

I tried that too

[rare talon]

You only need to find them in terms of x and y

No need for numerical value

If you're still insisting of numerical value, just use calculator on that

And it's pointless to have sin(17°) = x/y as the problem statement

[rugged moat]

Hmmmmmm

I will ask my teacher, what the question actually asks

[rare talon]

Okay

[rugged moat]

If I drew a triangle for the tanA tan2A problem will that work?

[rare talon]

I assume you know how to find cos(17°) and tan(17°) in terms of x and y

Anyway

[rugged moat]

Then,A=30

I know

[rare talon]

But tan(a) - tan(2a) = 1 doesn't have 30° as their solution

I don't think it helps

What kind of trig have you learned?

I want to do it with all the knowledge you know

Maybe not now, but tomorrow, since it's near midnight now

[rugged moat]

Give me a moment

I only know 3 identities:
Cos^2 +sin^2=1
1+tan^2=sec2

Cot^2+1=csc^2

Any luck?

[rare talon]

How you're supposed to do

Tan(A)-Tan(2A)=1
Sin^2(2A) + tan^2(2A)=?

If you only know that identity

No hope for me

[rugged moat]

Idk

I will ask my teacher tomorrow

In about 14 hrs, ping me if you want the solution

[brazen roost]

Wait

If tan a - tan 2a = 1

[dark sparrow]

F U CK I NG P A RE N TH E S E S

[brazen roost]

Then tan a > tan 2a

It's late at night

can't bother

[dark sparrow]

tan(a) can be greater than tan(2a) if a > 2a ;P

[brazen roost]

So

A is negative

But

Tan (A) is positive

[dark sparrow]

why should it be

[brazen roost]

Because Tan(A)-Tan(2A)=1

[dark sparrow]

so what

[brazen roost]

I dunno

[dark sparrow]

-6 - (-7) = 1

doesn't mean -6 is positive

[brazen roost]

Well

That works too

It's 1 30 in the morning and I can't sleep because of this problem

Curse you sheep

[gusty dagger]

What problem? Are we trying to find a solution for tan(a) - tan(2a) = 1?

Well, we can approximate it

Or

One solution

tan(a) = tan(2a) + 1
a = atan( tan(2a) + 1 )```

Now that we have `a` we can just substitute it back in for `a`, and we use some random number as a seat value for `a`

We'll use 2.4

```a = atan( tan(2a) + 1 )
a = atan( tan(2[atan( tan(2*2.4) + 1 )]) + 1 )
a = atan( tan(2[atan( tan(2[atan( tan(2*2.4) + 1 )]) + 1 )]) + 1 )```
etc, infinitely many times

It should either diverge or converge to a fixed point that satisfies `a = atan( tan(2a) + 1 )`

Lemme test it

.solve x = atan( tan(2*x) + 1 )

Q: x = atan(tan(2*x)+1)

=> x = (1.0991478671662067+0j)

A solution should be approximately 1.0991478671662...

.solve tan(1.0991478671662) - tan(2*1.0991478671662)

Q: x = tan(1.0991478671662)-tan(2*1.0991478671662)

=> x = (3.339380463511964+0j)

Crap

¯_(ツ)_/¯

My break is about to end so bye

[sand raven]

The Hardcore way xD

[minor quest]

What are they referring to as the segment area?

[shy bough]

I would interpret that to be the shaded area

[minor quest]

Alright thanks

@shy bough can you help me again?

Or anybody

How do you solve the hexagon area

[foggy oxide]

hello

here

a√3 = h
2

[shy bough]

took me a while to figure this out actually. but basically you can use the fact that a regular hexagon is made up of 6 equilateral triangles to derive a formula for the area of the hexagon

[shy bough]

oh prelude got you covered

[foggy oxide]

the trick it's what @shy bough said

use the fact that a regular hexagon is made up of 6 equilateral triangles

emphasis on the equilateral part

[minor quest]

Aww I yoloed it before you guys told me

[foggy oxide]

c00l d00d

[minor quest]

Ok .....

[shy bough]

sorry, didn't read till a while later and took me a while

[minor quest]

There's a formula using the apothem to get the sidelength on google that I tried

But it failed...

[shy bough]

that's probably what I did, using proper terms

side length works out to be 3

been over a decade since I did geometry :X

[minor quest]

When do you learn cosecant and etc

My geometry program is so weird

[shy bough]

I probably learned it ages ago but I mostly forgot it all and had to relearn it all when I started taking classes again

in precalc you use it a fair bit

and throughout calculus you'll use it a lot

[minor quest]

Well idk any other identity aside from the cos^2+sin^2=1 nothing about the cosecant thing not much about inverse trig and a bit about unit circle

[shy bough]

cosecant is just 1/sin

secant is just 1/cos

inverse trig functions just undo a trig function. which may or may not sound useful at all, but you're just finding the angle that creates the ratio you're given

you'll get a lot more practice with them in precalc and calc

wouldn't worry too much about it

[fringe basin]

I only met sec, cosec, cot this year i maths

for triangles

we just learnt sin is opp over hypo ect

[shy bough]

I much preferred the unit circle definition

made more sense and was easier for me to memorize

where sin(theta) = y/r, cos(theta) = x/r, and tan(theta) = y/x

and then sec and csc make more sense too, since they're just the inversions of those

csc(theta) = r/y, sec(theta) = r/x, and cot(theta) = x/y

[gusty dagger]

sec = 1/cos
csc = 1/sin
cot = 1/tan

Correct me if I'm wrong

[shy bough]

that's correct

[dull egret]

tan=sin/cos

[shy bough]

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

might help to think about it that way - did for me anyway

[foggy oxide]

well, unit circle is probably the best way to understand trig

trust me

[naive scarab]

it depends on what your view of the "real" definition is

you can take one or the other just as well, because they're the same

I like the unit circle better because it let's you define it naturally for all angles, it's easy to visualise, and it is more sensible as a definition for a function

[minor quest]

wat is t?

theta?

[naive scarab]

yeah

it's just the number you input to the function

[dark sparrow]

a.k.a. the argument of the function

😛

[grave echo]

Can anyone guide me

ABCD is a trapezium, AB is parallel with CD and AB+CD = BC. If the length of AD =12 then AB x CD is…

[dark sparrow]

hmm

this is surprisingly tricky

[rare talon]

I think I got it

Hello @grave echo

Do you know how to find BC in terms of AB and CD ?

[grave echo]

no

[rare talon]

Do you know pythagorean theorem?

[grave echo]

ye

[rare talon]

Try to make a line passing C perpendicular to AB

I name the intersection of that line and AB is E

[grave echo]

uh lemme try

hold on uh

[surreal bolt]

hmm if it is a number it isn't too hard 😃

[grave echo]

ye i got it

[rare talon]

Okay

[grave echo]

i made a line

[surreal bolt]

if it is an expression I might be stuck 😃

[rare talon]

BE = AB - CD right?

[grave echo]

uh wait

yes

[rare talon]

Good

Then we know on right triangle BCE:

BE^2 + CE^2 = BC^2

BE^2 is (AB-CD)^2

CE^2 is AD^2

[grave echo]

ye

[rare talon]

Okay try to expand it yourself 😛

[grave echo]

uh okay

[dark sparrow]

BC^2 = (AB+CD)^2 😉

[rare talon]

That's the 2nd equation that we'll construct

xd

[dark sparrow]

=tex (AB - CD)^2 + AD^2 = (AB + CD)^2

[grave echo]

CE^2 should be BC^2-BE^2

[dark sparrow]

i mean, yes

but CE = AD

[grave echo]

yea

[dark sparrow]

because AECD is a rectangle, by construction

[grave echo]

so 144 = bc^2 - be^2

what do i do next

[dark sparrow]

i feel like you're going in the wrong direction 😛

you know how to expand (x+y)^2, right?

[rare talon]

I'll take a leave, you're better at explaining things c:

[grave echo]

x^2+y^2?

[dark sparrow]

no

(x+y)^2 = (x+y)(x+y)

[grave echo]

eh ya

[dark sparrow]

= x^2 + xy + yx + y^2 = x^2 + 2xy + y^2

similarly, (x-y)^2 = x^2 - 2xy + y^2

[rare talon]

(you should make the table that sqrt2 did it for you again, if you forgot it)

[dark sparrow]

=tex AB^2 + CD^2 - 2AB \cdot CD + 144 = AB^2 + CD^2 + 2AB\cdot CD

[dark sparrow]

notice anything? @grave echo

[grave echo]

i need to disgest

[dark sparrow]

ok

[grave echo]

what am i looking for

[dark sparrow]

well

what does the problem ask you for?

[grave echo]

it asks me to do ABxCD

[dark sparrow]

yeah

AB * CD

anyway

if you look at that equation

i hope it's clear you can subtract away AB^2 + CD^2

and be left with 144 - 2 AB * CD = 2 AB * CD

do you follow?

[grave echo]

uh ya

eh

why 2AB

[dark sparrow]

(x+y)^2 = x^2 + y^2 + 2xy

[grave echo]

hmmm so it's

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

i think i get it

[dark sparrow]

yes

do you need further assistance?

[surreal bolt]

It checks with the example had in my head.

go Root2

[surreal bolt]

I formed a rectangle by rotating the trapezoid 180 degrees and putting the two pointy ends together. We wanted CB = AB + CD so I let AB and CD be the same. I had AB = 6, CD = 6, DA= 12. My final figure (if you count both "trapezoids") was a square. 6 X 6 = 36.

[sand raven]

sry

wrong chat

[foggy oxide]

the usual trick of those problems it's that you don't need to know directly the measures to find the answer

[rugged moat]

Apparently the tan(A)- tan(2a) problem I posted was actually tan(a)tan(2a)

#Sorry

[dark sparrow]

pff

right, tan(a)tan(2a) = 1, and what are we asked to find again?

[rugged moat]

tan(a)tan(2a)=1

Sin^2(2a) + tan^2(2a)

[dark sparrow]

aha

i got this

[rugged moat]

A=30

[dark sparrow]

yup, a = π/6

well

±π/6

sorry for the misshapen 2 in the second line lol

[foggy oxide]

someone here

[dark sparrow]

what do you need help with @foggy oxide

[fringe basin]

gotta love the whiteboard

[foggy oxide]

do you have a cool geo problem that u don't know the answer

[minor quest]

@dark sparrow can you further explain inceptionist's problem? I'm still trying to grasp how you would solve it.

[dark sparrow]

i've numbered the lines for convenience

@minor quest does this make sense?

if not, which line(s) would you like me to explain?

[minor quest]

shouldn't it just be 4AB = 144?

i divided the CD from both sides

[dark sparrow]

starting from what line

6?

(-2AB*CD + 144)/CD = -2AB + 144**/CD**

so no

dividing out by CD would not be helpful anyway since it's the product of AB and CD we want here

[minor quest]

Line 6 is -2ABxCD+144= 2ABxCD right?

[dark sparrow]

yes

[minor quest]

And I can rewrite it to 144-2ABxCD = 2ABxCD

So now I get rid of CD and turn it into 144- 2AB= 2AB

144= 4 AB

AB=36

[dark sparrow]

no

no you can't

if you divided both sides by CD you'd have 144**/CD** - 2AB = 2AB

=tex \frac{x+yz}{z} \neq x + y

[dark sparrow]

@minor quest

[minor quest]

How come the cd stayed with 144? @dark sparrow

[dark sparrow]

...

stop thinking of "getting rid of CD"

you're DIVIDING BOTH SIDES BY CD

=tex \frac{144 - 2AB \cdot CD}{CD} = \frac{2AB\cdot CD}{CD}

[dark sparrow]

THIS is what you're trying to do.

is that clear?

@minor quest

and NO, you CANNOT cancel out the CD's on the left hand side.
(x + yz)/z DOES NOT EQUAL x + y

[minor quest]

wait is it -2(ABCD) and 2(ABCD) or is it (-2AB * CD) and (2AB * CD)?

[rare talon]

The 2nd one

[minor quest]

@rare talon then how did it turned into 4AB*CD?

[rare talon]

Which part?

6 to 7?

-2AB.CD + 144 = 2AB.CD

Add 2AB.CD to both sides of the equation

If you're still confused, you can let AB = x and CD = y

It's just algebra

[minor quest]

shouldn't it be 4 AB * 2CD after you add 2AB * CD to both sides? @rare talon

[rare talon]

What?

[minor quest]

maybe im thinking too far into this..

[rare talon]

-2AB.CD + 144 = 2AB.CD
-2AB.CD + 144 + 2AB.CD = 2AB.CD + 2AB.CD
144 = 4AB.CD

Seriously

It's like you're doing this:
-3a + 1 = 5a
1 = 8a

[minor quest]

well you're not multiplying in that example

[rare talon]

Okay

-2xy + 1 = 3xy
1 = 5xy

It's literally the same

[minor quest]

isn't it more of 10 -2xy * da= 3xy* da?

[rare talon]

AB and CD are length of a side

You can let AB = x , CD = y

[minor quest]

i see

[rare talon]

Do you know like terms in algebra?

"Like terms" are terms that contain the same variables raised to the same power

[minor quest]

yes, yes i do.

[rare talon]

Even if I do something like:
1 - 3x^100 . y^99 . z^98 = 2x^100 . y^99 . z^98
It's the same as:
1 = 5x^100 . y^99 . z^98

I can perform this because the LHS and RHS has like terms, called x^100 . y^99 . z^98

[minor quest]

thank you for your help! @rare talon

[rare talon]

Okay

[dark sparrow]

@minor quest you do realize multiplication is associative, right?

x(yz) = (xy)z

[minor quest]

oh yes yes it is...

i just realized....

[cedar prawn]

How quaternions and rotation in 3space relate. SO PRETTY

@gusty dagger here it is

[gusty dagger]

Whoa

That's pretty awesome

I'm going to save that image

Thanks dude

[cedar prawn]

😃

[winged flicker]

when rotating a triangle, are side lengths not preserved?

[surreal bolt]

The side lengths are preserved.

[final prairie]

Rotations are isometries so preserved

[cedar prawn]

So I figured out the Gram-Schmidt Process thing I was doing, and got it working well

but I'm beginning to think it may be completely the wrong thing to do with what I'm trying to achieve

I mean, I've learned from it, so that's good

but it hasn't helped really

The whole thing I was trying to do was, define a plane of rotation on 4-space, and rotate a tesseract around the plane 3 degrees every 60th of a second, in order to produce an animation

Some instructions I found online said to use GS to make four orthonormal, orthogonal vectors and make a matrix out of them which would map the plane between the first two vectors and the origin to the XY plane - then rotate around XY which is quite simple - then map it all back to the plane it was on

and it sounds perfectly reasonable but when I actually applied it, the tesseract wasn't rotating, it was just wobbling

I've got my GS function working perfectly now, and as far as I know my function to make the rotation matrix as the guy said to is also working - at least, the points which ought to be invariant, definitely are - but it's not exactly rotating so much as jiggling

@dark sparrow I'm sorry to always be bothering you specifically - it seems like you're the one who always has to give help around here - but you are the only person who seems to be on right now who might maybe have some insight here.

[dark sparrow]

i've put the DO NOT DISTURB tag on myself. maybe, just maybe, it's there for a reason.

[cedar prawn]

Oh I'm sorry. I didn't see that - the little dot on your name is green

Never mind then

[brazen roost]

Are you still doing the thing from yesterday?

[cedar prawn]

Of course

[brazen roost]

Good luck on that m8

[cedar prawn]

I get the vague feeling that I annoy everyone here, sometimes

[brazen roost]

@dark sparrow

Bottom

That'd get the message across

[cedar prawn]

I wish I had someone to talk to about math who wouldn't 1. not know how to help 2. not care to help 3. be condescending or 4. give me a book to read instead of actually answering and who would actually enjoy talking to me about it

but so far no such persons appear to exist

But I guess everyone has that problem.

[brazen roost]

Hey

I don't know programming

So I can't help you in that regard

But if you want to talk about algebra, geometry, calculus, etc

Then the whole server is your friend

[cedar prawn]

In my experience the whole server is just a little bit hostile

I tend to ask questions and then get the feeling everyone thinks I'm stupid for asking them

Primarily because of the condescending aura of "what are you doing thinking about THAT, when you don't even know X Y and Z fundamentals"

people don't seem to understand the concept of "learn as you go along"

This is true on the Math subreddit as well. Lots of condescending, hostile people.

[brazen roost]

You're probably talking to the wrong person

[cedar prawn]

Most people I talk to here seem like that

[brazen roost]

There are lots of kind people here, you're just asking the wrong questions at the wrong time

And people just tend to assume that you get the basics down before asking more advanced and interesting questions

[upper karma]

There was one person 'hostile' to you and even then its quite an exageration

Imo

[brazen roost]

People tend to ridicule people that asks about integral calculus when they haven't even got polynomials down

[upper karma]

Someone who doesnt agree with your way of doing things and advice you to change it isnt hostile

[cedar prawn]

I can appreciate advice

but questions exist to be answered

and people always turn me towards books instead of just helping me solve my problem

[upper karma]

Because people have a way of doing things that you dont zgree with

[brazen roost]

Because we don't spoonfeed

[upper karma]

So they forward you to litterature so you can do as you will

[cedar prawn]

It's not spoonfeeding to explain something I would have learned from the book anyway

particularly as it is much faster than slogging through all the other things in the book that I don't need

[brazen roost]

If you can't even understand books, that's a sign you're not supposed to talk about it

[cedar prawn]

The fact is, I always understand everything I study eventually

[brazen roost]

Eventually

[upper karma]

See, this is how you always talk

Dont you feel you are condescending ?

[brazen roost]

But not at the moment you're asking it

[upper karma]

Just a bit ?

[cedar prawn]

I'm not being condescending

[brazen roost]

Sev

[cedar prawn]

Actually I sound like a pathetic loser making excuses for myself, if anything

[brazen roost]

what's he mad about again?

[upper karma]

Thats a way to see it as well

[cedar prawn]

Honestly I really have no idea what I'm mad about XD

I just feel like I don't belong here

[brazen roost]

From what I can see in the chat logs

[cedar prawn]

but that's true everywhere I go so. Maybe it's just my way of dealing with people

[upper karma]

You are making too much out of this

[brazen roost]

You don't seem to accept ideas that contradict your present belief

[cedar prawn]

That is true, I have a habit of being unwilling to accept criticism

[brazen roost]

Well

[cedar prawn]

The funny thing is I am perfectly aware of it

but... I haven't changed

[brazen roost]

That's your problem

Tackle that first

Then talk about math here

[cedar prawn]

Why do I always make a big thing out of every little thing and make myself look stupid 😦 Honestly that's the only thing that really bothers me here - the nagging certainty that everyone already hates my guts - it drives me to make it into a self-fulfilling prophecy by acting defensive. And then I try to explain myself (like I'm doing right now) when no one actually asked me to... gah!

[brazen roost]

Don't talk about it

Solve it

It's your problem

[cedar prawn]

I see why therapists make so much money

People like talking about things

but doing nothing

[brazen roost]

Exactly

[cedar prawn]

Maybe I should become a therapist

Wealth beyond imagining XD

[brazen roost]

You are aware of it, and I'm going to stop talking now

[upper karma]

No one hates you moon

To be honest most people dont care that much

[cedar prawn]

I should probably focus on equations

Math always makes me feel better

temporarily, anyway XD

[upper karma]

: ) maths is the best

I get that 😄

[cedar prawn]

Numbers and other mathematical objects are the best possible companions

They never say anything mean about you

and you can always predict what they will do

So trustworthy ❤

Plus, they are complex and mysterious and never bore you like real people always do

[upper karma]

^

[brazen roost]

In the illustration of quaternion rotation, where is the real line?

[cedar prawn]

Through the origin orthogonal to the three axes

it's the fourth dimension

I think

I didn't see the point of depicting it here

as the real part of quaternions is unnecessary for purely 3D rotations

[brazen roost]

I wonder what would happen to the point x = 1 after all that rotation

[cedar prawn]

x=1 isn't a point, in 3D it's a line, right?

wait

no a circle

on a sphere anyway

it would be a circle

[brazen roost]

No

Cartesian

[cedar prawn]

Ugh I am confused now

Oh it would be a plane in 3D then

[brazen roost]

No

A point

[cedar prawn]

x = 1

that isn't a point

[brazen roost]

Example of other point: x = 1 + i + j + k

[cedar prawn]

ohhh

I thought you meant

x coordinate

[brazen roost]

We're not talking about functions here

[cedar prawn]

functions? I thought you were referring to coordinates but you actually just mean the point (0,0,0,1) in 4-space

and you were referring to it in a quick way as a quaternion with only a real element

right?

[brazen roost]

Yes

But

You were referring to it as a plane

As if I'm talking about xyz space

But yeah

What would happen to x = 1?

[cedar prawn]

what would happen to it, when?

[brazen roost]

Rotated

[cedar prawn]

in what direction?

[brazen roost]

i.e. multiplied by i, j, k

And variations of the three

That'd be quite fascinating to visualize

[cedar prawn]

i * j * k * 1 = -1 so it would be reversed if you do all three

[brazen roost]

Not just that

I meant

i (-j) k

Or

(-i) j (-k)

etc

[cedar prawn]

Well the - is commutative in quaternion algebra so

there's really not much variation there

Wait

[brazen roost]

O really?

[cedar prawn]

I may be wrong

Let me think about it

[brazen roost]

You should explore it

[cedar prawn]

I feel like it is but I may be completely dead-ass wrong

[brazen roost]

And I should go to sleep soon because god it's late

Night

[cedar prawn]

It's morning here XD

night

[upper karma]

-1 is commutaive in the quaternion groups, its the generator of the center

[vapid kettle]

talking about unit groups?

or like

8 order quaternion group?

[upper karma]

Yes H_8

[vapid kettle]

I've never actually used quaternions in a geometric context 0.0

[upper karma]

Idk what the unit group of the quaternion ( field) is actually 🤔

[cedar prawn]

I can't imagine why anyone wouldn't

[vapid kettle]

hmmm

[cedar prawn]

Angles in 3D are confusing

quaternions make sense though

[vapid kettle]

well does it make sense?

unit gp in quaternion

it should but I like my commutativity

[cedar prawn]

Commutativity schmommutativity

[upper karma]

Idk if it does

[cedar prawn]

I like i² = j² = k² = ijk = -1

that's what I like

XD

[upper karma]

Matrices are great

[vapid kettle]

if you do some algebra you'll want your commutativity :3

[cedar prawn]

yes, everything is so much easier if you can use a matrix

[vapid kettle]

but place where quaternions came up was actually with

[cedar prawn]

at least with hypercomplex numbers

matrices make them way easier

[vapid kettle]

endomorphism algebras of some elliptic curves :0

which was pretty cool

[cedar prawn]

Oh dear. I don't even know what that means XD

[vapid kettle]

SEV is studying some elliptic curve stuffs

like the quaternion you talk about

I would call a Hamiltonian

[cedar prawn]

I'm guessing an elliptic curve is related to but not the same as an ellipse?

[vapid kettle]

it's related mmm

well do you know like

[upper karma]

Yeah ive seen that

[vapid kettle]

formula for the circumference of an ellipse?

[cedar prawn]

I can't remember it actually :3

but yeah go on

[vapid kettle]

well neither can I

[upper karma]

Its somewhat related but not as much as you would think

[vapid kettle]

because it's an elliptic integral so no one knows

or rather there isn't a nice answer

elliptic curves parameterize the calculation of the circumference of an ellipse

but they're essentially different things

[cedar prawn]

so, they are... approximations to ellipses which get closer and closer to the true circumference? or something?

[vapid kettle]

it's hard to describe elliptic curves in a nice way

I could say like

e.g. over the complex numbers

they're all tori

[cedar prawn]

I think a picture would be helpful here XD

or many pictures

[vapid kettle]

but a lot of what we do with elliptic curves is over

finite fields

which doesn't look like much of anything

[upper karma]

Finite fields are the best :3

[vapid kettle]

truth

[upper karma]

And local fields and number fields !

Number theory woo

[vapid kettle]

my algebraic number theory is weak so I don't even know what a local field is D: