or trig identity
#help-36
terse crypt
soft zealotBOT
k
terse crypt
thx
rotund iris
ari skin chung
arctic fiber
Would creating a triangle that has angle A and 90° work using that ratio?
terse crypt
it would
cosA = adjacent/hypotenuse
compare that to the fraction
and then pythagorus
arctic fiber
Is the tanA √20/√5 then?
terse crypt
,calc sqrt(5 - 1)
soft zealotBOT
Result:
2
ye
lucid nymph
you really simplified by de-rationalising the denominator?
terse crypt
you really simplified by de-rationalising the denominator?
yes
i love 1 in the numerator
[ \sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}} 😻 ]
soft zealotBOT
k
maiden vessel
you really simplified by de-rationalising the denominator?
it looks better like that
rotund iris
do u must need to use english here?
terse crypt
nah
meager mirage
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tawdry eagle
<@&286206848099549185> pls help
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<@&286206848099549185> pls help
storm haven
!15m
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worthy wren
factorize a^3+1
jagged flare
let 2024 be a variable call it a
left acorn
zenith cedar
!occupied
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tawdry eagle
tell me what you get after factoring
sqrt of 2024(2024^2+1/2025)
worthy wren
no, just factorize the expression a^3 + 1
final saddleBOT
@tawdry eagle Has your question been resolved?
brave pivot
I am getting $4a^2+a-1=0$ where $a=sin^2(x)$; that gives a weird ahh answer. can anybody check...
final saddleBOT
I am getting $4a^2+a-1=0$ where $a=sin^2(x)$; that gives a weird ahh answer. can...
soft zealotBOT
Double_mytrouble
loud sundial
No
cinder spear
last river
How did you simplify the equation? Can you share the process?
tired walrus
#discussion or #chill please.
tired walrus
I am getting $4a^2+a-1=0$ where $a=sin^2(x)$; that gives a weird ahh answer. can...
show work for how you did that 
brave pivot
show work for how you did that <:thonk:406575732563902485>
used product to sum
all became either x or x/2
tired walrus
i said show not tell
brave pivot
ok
tired walrus
i want to rule out any possibility of stupid arithmetic fuckup on your part
brave pivot
tired walrus
line 2, what does this say?
tired walrus
wdym changed
last river
In step 2, sin(x) + sin(x/2) is 2sin(x)cos(x/2)
And not 2sin(x)sin(x/2)
tired walrus
cos(5x/2) is not the same thing as cos(x/2) mate
brave pivot
Value wise they are same nah...
brave pivot
Bruh
last cape
You are correct.
brave pivot
You are not getting it
tired walrus
fym im not getting it
where are you getting it from that cos(5x/2) can be just replaced out of the blue with cos(x/2)
brave pivot
They aren't same on graph (Obv) but take it in a unit circle
tired walrus
you cannot replace cos(5x/2) with cos(x/2) in an equation
brave pivot
where are _you_ getting it from that cos(5x/2) can be just replaced out of the b...
Bruh, don't you know cos(2π+x)=cosx
brave pivot
you cannot replace cos(5x/2) with cos(x/2) in an equation
I can... I will just take period back
brave pivot
You are pmo
tired walrus
x is not pi
brave pivot
Leave it
tired walrus
please do not call me "bro".
brave pivot
I did many questions using that. It worked before. Our teacher uses it
tired walrus
please edit the word "bro" out of your message.
brave pivot
Sure
last river
But that is wrong, you know
tired walrus
I did many questions using that. It worked before. Our teacher uses it
can you show me at least one where cos(5x/2) (not cos(5pi/2) or any other expression) was replaced with cos(x/2) (not cos(pi/2) or anything else)?
last cape
I am not sure.
tired walrus
You are pmo
i'm pissing you off by doing what exactly? pointing out an instance of you being wrong and/or misremembering some tricks from class?
final saddleBOT
split nebula
idk how you're arguing with desmos lol
tired walrus
idk how you're arguing with desmos lol
ive seen it happen sometimes that ppl dont want to admit their own wrongness
brave pivot
I am asking in our group, if I am wrong, I will admit
last cape
Me too.
split nebula
yeah that does happen
split nebula
I am asking in our group, if I am wrong, I will admit
do you know what desmos is?
brave pivot
do you know what desmos is?
No...
split nebula
it graphs functions
split nebula
so these are graphed and you can't exactly debate this
brave pivot
ohh ok, lemme look it up
last cape
last river
I did many questions using that. It worked before. Our teacher uses it
Ok, can you prove it? Can you give a legit proof that cos(5x/2) = cos(x/2) generally?
brave pivot
Lemme see
last river
And not just claim that, "That's how it works, I've used it many times"
tired walrus
tired walrus
I am saying it worked
you've yet to show us any fully worked solution in which it worked
brave pivot
And not just claim that, "That's how it works, I've used it many times"
Shit, I am wrong... in the previous question as putting x I was getting sinx=0. so, it worked as x=0 was solution
My bad
tired walrus
told ya 
near reef
unbelievable argument happening right now. plug in x=pi and it's immediately apparent that it's false because the angle is in different quadrants
last cape
Me too.
tired walrus
do we want to reopen
near reef
I have made many silly mistakes before as well, but you were genuinely disrespectful about it imo
brave pivot
do we want to reopen
nah, sir said, he will discuss it in class
Class is in a hour so...
near reef
even if you are 100% sure that you're right, it's always important to be civil and explain your reasoning in a non-condescending way
last river
Now proceed with
$cos(\frac{5x}{2})cos(\frac{x}{2}) = sin(x)cos(\frac{x}{2})$
soft zealotBOT
PRAKHAR
tired walrus
\cos and \sin
last river
`\cos` and `\sin`
Oh, thanks
last cape
Now proceed with
cos$(\frac{5x}{2})$cos$(\frac{x}{2}) = $sin$(x)$cos$(\frac{x}{2})$
soft zealotBOT
Genius
last cape
No need for \cos and \sin
tired walrus
Now proceed with
cos$(\frac{5x}{2})$cos$(\frac{x}{2}) = $sin$(x)$cos$(\frac{x}{...
this is atrocious.
final saddleBOT
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final saddleBOT
sullen gale
!noping
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final saddleBOT
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latent dragon
You’re supposed to ping members after 15 minutes of your question being unanswered. I don’t see a question.
sullen gale
also, please post your question
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brazen gazelle
.close
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strong flax
help
strong flax
simplifying square roots is soo confusing
atomic moon
How so
strong flax
well i think i got a formula / trick
to skip all the algebra
but when its numbers like these its crazy
sage osprey
strong flax
so sapien
ill tell you my secret technique
ive figured something out
so
basically you can skip all the algebra
tired walrus
is the 80b^2 meant to all go under the root symbol
strong flax
so
its kind of hard to explain
if the number is clean like 80
we have 16 * 5
pick like a perfect square that will equal to 80
tired walrus
im asking about your handwriting: in the equation you wrote here, is the 80b^2 meant to all be under the root
yes or no
atomic moon
its kind of hard to explain
Do it to the question then
strong flax
oh yea
tired walrus
in that case:
strong flax
wait
tired walrus
your root symbols should extend like this and actually physically cover all the shit that goes under them
tired walrus
hold on, i'm trying to give you a concrete pointer for fixing your handwriting here.
strong flax
OHHH
I FIGURED IT OUT
💁
i figured out the secret technique to skip all the algebra
but it only works for clean numbres
it only works for nice numbres i guess
ok
u know what
im just gonna skip this
and come back to it later
y'all I GOTTA TAKE A BREAK
magic coyote
K
strong flax
.close
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magic coyote
!done
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tough pollen
hi
strong flax
hiiiiii
magic coyote
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winged magnet
Is 32 the only power of two that is a part of this function? $$P\left(x\right)=\sum_{n=1}^{x}n^{n}$$
final saddleBOT
Is 32 the only power of two that is a part of this function? $$P\left(x\right)=\...
soft zealotBOT
Vanouper
winged magnet
P(1) = 1
P(2) = 5
P(3) = 32
P(4) = 288
P(5) = 3413
P(6) = 50069
P(7) = 873612
...
32 seems to be the only one. Any reason why?
Oh wait, excluding P(1)... but that feels trivial.
Feels trivial because any number to the zero is one.
final saddleBOT
@winged magnet Has your question been resolved?
winged magnet
<@&286206848099549185>
kindred sun
P(1)
thats a power of 2
oh wait
i didnt read th enext one
yea its the only one
winged magnet
1 is a trivial case. (x^0=1)
winged magnet
yea its the only one
How can it be proven though? That 32 is the only power of two that is a part of the function?
kindred sun
for x≥4:
P(x) quickly becomes odd making it not a power of 2
The modular residues of P(x) don’t match any power of 2
The growth of x^x ensures P(x) falls between powers of 2, never equaling one
winged magnet
for x≥4:
> P(x) quickly becomes odd making it not a power of 2
> The modular res...
That explains it. Thanks.
.close
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fiery pine
for x≥4:
> P(x) quickly becomes odd making it not a power of 2
> The modular res...
how do you prove any of those 3 things?
final saddleBOT
how do you prove any of those 3 things?
fiery pine
no?
fiery pine
it cycles even even odd odd
fiery pine
5,
32,
288,
3413,
50069,
873612,
17650828,
405071317,
10405071317,
295716741928,
9211817190184,
312086923782437,
11424093749340453,
kindred sun
The modular residues of P(x) don’t match powers of 2
use mod 3 or 4
it can bring up inconsitiensies
So from
x=4 onward, P(x)mod3 starts hitting 0 proves that it is impossible for any power of 2 except 0
proved: Modulo residues don’t match powers of 2
fiery pine
wat
kindred sun
also because n^n growth is just too big to stay same speed as any poewr of 2 from n≥5
so basically modulo residue fails
p(x) becomes odd
and it just grows too fast in general
fiery pine
P(x)mod3 starts hitting 0
so what, it is also 1 or 2 mod 3 infinitely often
P(x) does not become odd eventually omg
fiery pine
5,
32,
288,
3413,
50069,
873612,
17650828,
405071317,
10405071317,
2957...
like did you look at this?
the sequence 0,1,2,3,4,5,6,7,8,... mod 3 also starts hitting 0. but it does have powers of 2 in it
final saddleBOT
safe arch
Guys what’s hardest math question?
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Available help channel!
drowsy epoch
,The 1 million dollar questions.
final saddleBOT
,The 1 million dollar questions.
drowsy epoch
.close
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blissful zenith
Wait you can earn a million?? Studying this?
drowsy epoch
Wait you can earn a million?? Studying this?
I was referring to the millenia problems.
blissful zenith
I was referring to the millenia problems.
I didn’t know them I just searched 😢
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safe arch
Solve the inequality 4/(|x|+2) > x^2 +3x +1
final saddleBOT
Solve the inequality 4/(|x|+2) > x^2 +3x +1
severe canyon
!status
final saddleBOT
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2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
old quarry
i would suggest doing it graphically
scarlet sequoia
Solve the inequality 4/(|x|+2) > x^2 +3x +1
if you don't know how to solve cubics, I suggest you content yourself with approximate graphic values
old quarry
yes and the solution to this inequality is not very nice
inner acorn
I would suggest case aalysis instead
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untold elk
.
strong flax
HELP
old quarry
w what
sour zealot
do you have a question?
severe verge
they've done this twice now
magic coyote
.close
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magic coyote
Reopen if u have a question
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devout meadow
A juggler alternately throws three oranges that rise 1.8m above his hands. It takes him 0.3s to move one orange from one hand to the other. When one orange reaches its maximum height, where are the other two? Assume they are evenly spaced in time. Overlay the graphs of the vertical position versus time of the three oranges to verify the answer to the question.
final saddleBOT
A juggler alternately throws three oranges that rise 1.8m above his hands. It ta...
devout meadow
please
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@devout meadow Has your question been resolved?
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solar lichen
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solar lichen
can i get help for d im confused
solar lichen
how do it tbh
tired walrus
245° is represented by one of those arrows they drew for you in there
solar lichen
245° is represented by one of those arrows they drew for you in there
how do i figure out which??
tired walrus
which quadrant is 245° in?
solar lichen
3??
tired walrus
was that a guess or was that sheer uncertainty
tired walrus
what range of angles is the 3rd quadrant
solar lichen
65
tired walrus
...
solar lichen
uhm
tired walrus
the third quadrant contains the angles from what to what?
solar lichen
180-270
solar lichen
yes
tired walrus
note also 245=180+65
solar lichen
ohh
tired walrus
which is the whole reason why that arrow in the 3rd quadrant represents 245 in the first place
tired walrus
you would read off the y coordinate of the point where that ray and the unit circle intersect
which, yes, is -0.9
tired walrus
you once again have a whole-ass arrow representing it
this time in the fourth quadrant
-40° means you walk 40° clockwise
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solar pelican
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solar pelican
Determine natural numbers n for which
Can you give me a hint with which to begin?
Ive only said that x-1, x-2 ... x-n >= 0
jagged flare
my firsr thought is maybe try induction to prove that if we fix n, then if it holds for x=2n then it holds for all x>=2n?
solar pelican
Ill try that
jagged flare
actually idt this works
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@solar pelican Has your question been resolved?
jagged flare
diffrent idea: let $x=2n+a$, let $f(a)=\sum_{i=n+a}^{2n+a-1}\sqrt{i}-a$, so basically it becomes for all $a\geq 0$ then $f(a)<2n$, then we can maybe find what $a$ is when $f(a)$ is maximized
soft zealotBOT
skissue.in.a.teacup
jagged flare
idk maybe im juat yapping and delulu
vital surge
am-gm gets rid of 4
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@solar pelican Has your question been resolved?
severe verge
@solar pelican got the idea
solar pelican
tell me
severe verge
so use AM-GM on the left side we have
LHS > n.[(x-1)(x-2)...(x-n)]^(1/2n) > n.√(x-n)
Which means x>n√(x-n)
from this, prove that this is untrue for n>=5
whole halo
tested, this is also untrue for n >= 4, since you need x >= 12.8492 instead of x >= 8
additionally,
severe verge
for n=4 if you're not allowed to use a calculator...
x>4[(x-1)(x-2)(x-3)(x-4)]^1/8
x^4 > 256(x-2)(x-4)
Plug in x=8 and it'll be false
You might have to prove that the inequality is true for n = 1,2 and 3 as well
severe verge
yeah
final saddleBOT
@solar pelican Has your question been resolved?
severe verge
@solar pelican are you stuck anywhere?
severe verge
oh lol
severe verge
I just solved like 60% of the problem
solar pelican
<@848592586134519841> are you stuck anywhere?
I just solved it
Thanks again! I wouldve never thought of am-gm honestly
severe verge
ye at first I thought of using Minkovsky or sth
until I realize that using loose inequalities works really well for some reason
solar pelican
i spun around with square root conditions and achieved nothing
thanks again, ill close this now
have a good day
!
.close
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severe verge
np
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magic coyote
@undone tiger how did u get diffb??
final saddleBOT
<@1340556085467287603> how did u get diffb??
undone tiger
I simplified first function as 1+4x+x^2/2
Derivative will he 4+2=6
And second equation derivative is 5
magic coyote
Wut
warm python
!xy
final saddleBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
magic coyote
How did u simplify $\int_0^x 5+|1-t| dt$ to that
soft zealotBOT
magic coyote
I simplified first function as 1+4x+x^2/2
Derivative will he 4+2=6
And secon...
And even by this it means they're not differentiable 😭
What fr
undone tiger
How did u simplify $\int_0^x 5+|1-t| dt$ to that
I broke it down 0 to 1 and 1 to x?
and integrated
Is it necessary to be equal to differentiable?
magic coyote
Yes 
magic coyote
I read somewhere that it's not necessary
final saddleBOT
serene panther
Is it necessary to be equal to differentiable?
for what
undone tiger
To differentible at some point
serene panther
well RHD = LHD always for differentiability at any point
undone tiger
Koi point par function agar differentible hoga agar uske LH derivative and right derivative ka exist hona chahiye
Equal hona v compulsory h kya?
undone tiger
well RHD = LHD always for differentiability at any point
I doubt
serene panther
yes
yes
obviousl
y
take |x|
LHD not equal toRHD
but they still exist
and |x| is not differentiable at 0
differentiability implies the function must be smooth
no cusps, sharp peaks
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teal swallow
$\text{Let } P(x) \text{ be a polynomial of degree } n \geq 1 \text{ with } n \text{ distinct real roots.} \
\text{Prove that } Q(x) = P'(x) - 2x P(x) \text{ has exactly } n+1 \text{ distinct real roots.}$
final saddleBOT
$\text{Let } P(x) \text{ be a polynomial of degree } n \geq 1 \text{ with } n \t...
soft zealotBOT
TargetVN
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What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
im typing
give me time
This has to be the weirdest problem I have ever encountered.
- First of all, why is this in Calculus I? Yes, I know, there's that P'(x), but this problem seems so innocent that it looks like it should not be solved using calculus
- Now, if I let Q(x) = 0, it seems like it's a very normal ODE problem, but that doesn't look like it'll help anything, at least that's what I thought.
Also, where the hell am I supposed to start???
dusty quarry
this sounds like a job for the polynomial factor and remainder theorems
loud sundial
Consider the derivative of ||e^(-x^2) P(x)||
dusty quarry
this actually sounds quite simple
dusty quarry
this sounds like a job for the polynomial factor and remainder theorems
but i think by mentioning this, do you now have an idea? @teal swallow
teal swallow
still no idea
teal swallow
Consider the derivative of ||e^(-x^2) P(x)||
i can probably work around this tho
loud sundial
but i think by mentioning this, do you now have an idea? <@373752699382136833>
What’s your solution? Because mine is just using what I said above followed by ||some rolle’s theorem and behavior at infinity|| and type arguments
teal swallow
since its basically G'(x)=0
loud sundial
What’s your solution? Because mine is just using what I said above followed by |...
Or are you just spitballing
loud sundial
since its basically G'(x)=0
Yeah basically - the rest is fairly straightforward after that
teal swallow
it is?
hold on
ah, so rolle's theorem for every (x_k, x_{k+1}) (x_i is root of P(x) = 0)
dusty quarry
What’s your solution? Because mine is just using what I said above followed by |...
i didn't know about this, but my initial instinct was this (probably doesn't apply for all polynomials, but i'll word my instinct anyway)
P(x) has n roots.
P'(x) should then have n-1 roots (degree of first derivative is 1 less than degree of original function
xP(x) should then have n+1 roots (because of the extra x being multiplied in)
addition / subtraction operations involving two polynomials, one of degree n-1 and the other of degree n+1, should create a polynomial of degree n+1
loud sundial
i didn't know about this, but my initial instinct was this (probably doesn't app...
Yes but you need to argue that all of those roots are real and distinct
Which is the main part of the question
dusty quarry
ok that's a fair point i didn't consider
loud sundial
Finding the degree of the polynomial is mainly a trivial/fleeting thought
You use it at the end but it’s not particularly interesting
teal swallow
ah, so rolle's theorem for every (x\_k, x\_{k+1}) (x_i is root of P(x) = 0)
hold on, this covers only n-1 roots
loud sundial
mhm
teal swallow
where are the other 2
loud sundial
Think about ||splitting up the real number line into intervals||
What have you not considered yet
teal swallow
oh
im stupid
(-inf; x_1) and (x_n; +inf)
Both those interval have 2 more
Alright, q.e.d
Thanks for help
❤️
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loud sundial
Thanks for help
teal swallow
pretty fun problem tbh
loud sundial
Yeah it’s a cute one
Nice way to spend the first part of my train ride lol
Oh good timing the second train just arrived
sullen gale
<@&268886789983436800>
spam opening channels
and then deleting the first message
@short radish <--
barren hound
it was tripping the automod but not actually muting 
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uneven yacht
z^4 = -4, find all solutions
final saddleBOT
z^4 = -4, find all solutions
edgy mauve
Is that possible
exotic shale
z^4 = -4, find all solutions
Hay
serene panther
Do you know abt roots of unity
tired walrus
Is that possible
in the complex numbers, it is
solar crest
z^4 = -4, find all solutions
is z some complex stuff?
exotic shale
z^4 = -4, find all solutions
So one way to do this is write -4 in exponential form
solar crest
in the complex numbers, it is
oh
serene panther
I mean its z so prolly complex
exotic shale
Using Eulers identity
uneven yacht
Do you know abt roots of unity
+1
exotic shale
You can write it as 4e^iπ
exotic shale
Then take the 4th root of that
solar crest
is roots of unity
🤩 or 🥀
idk if i should be excited for when i learn it or scared
uneven yacht
Do you know abt roots of unity
u r confusing me why ur asking roots of unity
serene panther
u r confusing me why ur asking roots of unity
Because its related to solving the problem
What you said are not the entire roots of unity
exotic shale
Actually wait i made a mistake
serene panther
The eqn z^n = 1 has n roots
exotic shale
z^4 = 4e^i(2kπ +π)
serene panther
All of the form e^(i * 2pi/n)
exotic shale
z^4 = 4e^i(2kπ +π)
Then
z= 4th√4 e^((2κπ +π)/4)
For k from 0 to 3 natural numbers
So for κ ={0,1,2,3}
Can also do it in cis(θ) form but they are the same with moivre theorem
Not sure if i spelled name correctly
uneven yacht
What you said are not the entire roots of unity
bro, are u asking square root of 1 or some equation?
how can a number have roots? only equation can have roots
solar crest
how can a number have roots? only equation can have roots
but it’s complex
uneven yacht
1 is a constant
exotic shale
how can a number have roots? only equation can have roots
Think he is talking about the plane representation
serene panther
1 is a constant
Sigh
exotic shale
Its a vector rotating splitting the 360° in n equal angles
And vectors of same length from 0 point to all roots with nθ angle
exotic shale
is roots of unity
🤩 or 🥀
idk if i should be excited for when i learn it or s...
All you really need to do is swap forms
Just write stuff using Eulers identity either trigonometric or exponential
exotic shale
Nah not really
I mean unless you are going in stuff like complex analysis etc
The basics you learn in complex numbers aren't that advanced
You just have to think that imaginary numbers cover a plane not a line
Its good to watch yt videos
uneven yacht
How do i solve this x^2 - y^2 + 2xyi = +-2i it's getting messed up
solar crest
Its good to watch yt videos
which would u recommend to learn early?
the organic chem tutor?
im gna go on khan academy and learn it earlier there
solar crest
oh?
interesting
exotic shale
Their goal is to get views so they just make it so that anyone can get a rough understanding
Id just look for videos analysing and explaining rigorously as much as possible atleast
uneven yacht
How do i solve this x^2 - y^2 + 2xyi = +-2i it's getting messed up
ok wait i actully got something nice
exotic shale
How do i solve this x^2 - y^2 + 2xyi = +-2i it's getting messed up
x,y are what
in R or C
uneven yacht
x = -yi +- sqrt(2i)
uneven yacht
x,y are what
z = x + yi; x,y belongs to Real
exotic shale
Ok so x,y are real numbers
Ok then you just match real and imaginary parts
x^2 -y^2 = 0
And 2xyi = +-2i
And solve the system
It should be all combinations of 1 and -1 for each x and y
uneven yacht
Ok then you just match real and imaginary parts
ah, thanks
i forget to match parts and go for quadratic formula idk why D:
ok got the answers
thanks
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old drum
Pls solve this
final saddleBOT
Pls solve this
dusty quarry
unless i'm dreaming or have lost my eyesight, i don't think you've sent the question
dusty quarry
what have you tried?
severe verge
dusty quarry
wait this is a sticker here? got it
severe verge
heheh
dusty quarry
wait so you know how to solve but have tried nothing? am i misunderstanding or
dusty quarry
looks like it
old drum
Yeah
magic coyote
final saddleBOT
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thin cloud
Given a tetrahedron. How many tetrahedrons can we form using the vertices, the midpoint of lines, the centroids of the faces, and the centroid of the given tetrahedron?
final saddleBOT
Given a tetrahedron. How many tetrahedrons can we form using the vertices, the m...
thin cloud
I had a hard time counting invalid cases when 4 points are in the same plane
severe verge
the centroids of the sides???
tired walrus
the centroids of the faces i think is what's meant
tired walrus
what's your native language?
severe verge
Vietnam
tired walrus
hmm ok so
my thinking is that if 4 points are coplanar then the plane they determine is unique
bc we don't have any four collinear pts in the picture
severe verge
I know him
solar crest
ah
severe verge
if we have midpoints only then we have 4 maybe?
tired walrus
and we can also have them share a bisecting plane that runs through one edge and the midpoint of its opposite
lemme take a pic
thin cloud
I know that each face has 7 coplanar points
tired walrus
there are 6 such planes, one for each possible contained edge
however we can also have a different kind of plane that goes through the midpoints of a loop of 4 edges
thin cloud
there are 6 such planes, one for each possible contained edge
I also thought of it.
thin cloud
however we can also have a different kind of plane that goes through the midpoin...
Ohhhh, that's what i missed
tired walrus
i think these only contain the centroid of the whole tetrahedron
and no other points we care about
there are 3 such planes, one for each pair of edges they're parallel to
thin cloud
So we have 4 planes for the faces, 6 planes for edge and 3 planes for what you just said
tired walrus
yes
though notice that the face and red planes contain 7 pts each while the blue ones contain only 5
this affects counting
thin cloud
Still missing something, I do 15C4-7•7C4-3•5C4=1000
It's a multiple choice and it doesn't have 1000 as an answer
severe verge
@thin cloud Má a tưởng đếm hình tứ diện đều
thin cloud
Okay we forgot about plane that include 1 vectice, 2 centroids of two faces and two midpoints
tired walrus
these go through... a vertex and two of the opposite face's side midpts
severe verge
Still missing something, I do 15C4-7•7C4-3•5C4=1000
Sao lại trừ 7C4 7 lần
thin cloud
Sao lại trừ 7C4 7 lần
10 lần,ghi lộn đó 😂
severe verge
để a nhìn lại
severe verge
15 đâu ra
thin cloud
Mỗi đỉnh ứng với 3 cặp đường trung bình của tam giác đối diện và nó cũng đi qua trọng tâm luôn
thin cloud
I figured it out, it's 15C4 - 10•7C4 - 15•C4=940
severe verge
6 thôi mà????
thin cloud
15 xd
Em check đáp án đúng rr
Mỗi đỉnh có 3 mặt phẳng đi qua đỉnh đó và đi qua 1 trong đường trung bình của mặt đối diện
Và mặt phẳng đó cũng đi qua trọng tâm 2 mặt bên luôn
thin cloud
these go through... a vertex and two of the opposite face's side midpts
I figured it out Ann, we have 12 planes like that and each plane have 5 points. Thanks for your help
thin cloud
lấy tử đâu bài bỏ con thế
Đề ôn thi TSA 😂
thin cloud
Thật à
severe verge
uk
thin cloud
Là khó hơn hay dễ hơn
severe verge
Độ khó quá cao so với TSA
thin cloud
Ohhhh, XD okayyy, nghe có động lực hẳn
thin cloud
thôi m 90 TSA đi là vừa
Thanks anh xd
severe verge
a có 72 nói làm j
thin cloud
Muộn lắm rồi
thin cloud
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gusty sedge
i have a question regarding those who are studying math at university. what math should i study to make myself ready?
final saddleBOT
i have a question regarding those who are studying math at university. what math...
vital crag
i have a question regarding those who are studying math at university. what math...
Help channels are for specific math problems or questions
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restive snow
final saddleBOT
restive snow
I am working on the first one right now
Since AB=BE, triangle ABE is isosceles, which means that angle AEB = BAE = 46°
This makes angle ABE 180-92 = 88°
This makes angle EBC = 180-88=92
I dont know where to go frmo here
severe verge
yep
severe verge
EBCD is sth special
restive snow
trapeziod?
severe verge
no
severe verge
since the quadrilateral is cyclic you can imagine E,B,D and C all lying on a circle
now what is angle EBD equal to
restive snow
yeah
severe verge
precisely
severe verge
actually do you know the ways to prove that a quadrilateral is cyclic?
restive snow
if the opposite angles are supplementray?
severe verge
alr
so there are 2 sub categories
proofs based on angles and proofs based on length
restive snow
mhm
severe verge
let's start with angles
restive snow
okay
severe verge
so as you can see A and B lie on the same side with respect to the chord CD
if CAD = CBD then ABCD is a cyclic quad
restive snow
mhm
severe verge
consequently if the angle is equal to the exterior angle of the opposite side then it is cyclic
restive snow
would this mean that if angle B is equal to the exterior angle of C rhe. its cylic
severe verge
yep
restive snow
ooh
hasty sleet
has someone worked on the square?
restive snow
has someone worked on the square?
not yet im gonna do that one afterwards
severe verge
wait did the image send
restive snow
if IA.ID = IB.IC or KA.KB = KC.KD then cyclic quad
are you multiplying ia and id
hasty sleet
oh okay
restive snow
yea
severe verge
are you multiplying ia and id
yep
severe verge
yep
restive snow
i see
severe verge
not in this case tho
now going back to the square one
there's an angle you can easily calculate
since ABCD is a square, what angle is BDC?
restive snow
hmmm
severe verge
there are 2 angles that are 45°
one of the mistakes I see when people look at angles is they pay too much attention to the end points
like you know angle BDC = 45°
restive snow
yeah
severe verge
but staring at B and C doesn't help much in this casr
what do I see at B and C? Nothing. Because the angle is at D
severe verge
alr what is it
severe verge
there are 2 angles that are 45
severe verge
AEG?
severe verge
correct
restive snow
yea
severe verge
you see angle at A and D = 45°
restive snow
mhm
restive snow
wait is it 90
severe verge
Yep
restive snow
wait so it EGCF also cyclic
restive snow
45?
severe verge
nice job
restive snow
hurrayy
severe verge
it's amazing
like when you look at the diagram at first, E,F and G are just 3 random points
and you managed to find the angles on them
we also didnt use the 31° angle at all
restive snow
its pretty cool
yeahh
woah
tysm for your help
you helped me yesterday too
im gonna close it now
thank you so mcuh
.close
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severe verge
np
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quartz wind
A satellite of m is rotating around a planet of M on a circular orbit of R. Due to space dust, it gradually decreases the radius of the orbit to R/2. Assume that the satellite performs uniform circular motion at any moment. How much energy is dissipated because of space dust?
final saddleBOT
A satellite of m is rotating around a planet of M on a circular orbit of R. Due ...
magic coyote
Do you know the formula for TE
quartz wind
Do you know the formula for TE
like Mechanical energy?
magic coyote
yes, total mech energy
quartz wind
I think its kinetic plus potential right
magic coyote
yes, that's correct, but have you derived it for satellites specifically?
quartz wind
so your asking if i had found the kinetic and potential energies for the satellite?
magic coyote
sure
quartz wind
for both initial and final positions?
magic coyote
yes
quartz wind
yea
quartz wind
k(ini) = mv(ini)^2/2
k(fin) = mv(fin)^2/2
UG(ini) = -GMm/r
UG(fin)= -2GMm/r
magic coyote
ok, sure, but whats the initial and final speeds
quartz wind
ok, sure, but whats the initial and final speeds
untold
magic coyote
yes, but u do know how to find them?
quiet zinc
untold
are you sure the term rotating is used
i mean it feels like orbiting must be mentioned
quartz wind
i mean it feels like orbiting must be mentioned
yea i think its just a typo
its rotating but it prob means orbiting
magic coyote
i mean it feels like orbiting must be mentioned
same thing kek, pri obviously gravitation qn
quiet zinc
same thing kek, pri obviously gravitation qn
yeah fair , otherwise would be insufficient data
magic coyote
magic coyote
yes, but u do know how to find them?
?
quartz wind
yes, but u do know how to find them?
like isolate them?
magic coyote
$v_{orbit}=?$
soft zealotBOT
magic coyote
k(ini) = mv(ini)^2/2
k(fin) = mv(fin)^2/2
UG(ini) = -GMm/r
UG(fin)= -2GMm/r
actualy your U values are wrong
it says that it's rotating at a distance of R and R/2 from the surface
so the radius of rotation is the radius of the earth+that distance
quartz wind
Think of r as the center of mass
magic coyote
?
quiet zinc
it says that it's rotating _at_ a distance of R and R/2 from the surface
no?
wait
its orbiting at r yeah
he wrote d , should have been r tho
quiet zinc
Think of r as the center of mass
sorry?
magic coyote
he wrote d , should have been r tho
how can it rotate at r/2
r/2 would mean inside the earth
it's 3r/2
quiet zinc
r/2 would mean inside the earth
we dont know that?
bro radius is not r
you assumed the radius of earth is r
unconciously
quartz wind
R is not radius of earth
quiet zinc
yeah focus on what executor was asking tho
magic coyote
I thought energy wasn't conserved
energy is conserved
and idk wym by bringing com into it
quartz wind
does space dust not do work?
quiet zinc
lets just remove everything from the question
and lets assume this
A spherical mass (M) with Radius R , a small spherical mass (m) orbiting around the spherical mass at a distance d from the centre of the spherical mass M
(M>>>>>m)
whats the velocity at which spherical mass is traveling at to maintain its orbit radius
quartz wind
so were isolating v?
magic coyote
Minimum velocity, mind you
magic coyote
so were isolating v?
from the energy conservation equation you'll come up with by considering the earth's center as U=inf, yes
quartz wind
Wait so we are not going to consider space dust a force right
magic coyote
lets just remove everything from the question
and lets assume this
A spherical ...
ans this first
quartz wind
v(ini) = sqrt(2GM/R)
quiet zinc
v(ini) = sqrt(2GM/R)
define R real quick
is this the distance between the object and the satelite
quartz wind
define R real quick
distance from center of mass to orbit
quiet zinc
distance from center of mass to orbit
nah thats not it . you are confusing escape velocity with weird orbiting velocity
magic coyote
v(ini) = sqrt(2GM/R)
That's the escape velocity
quiet zinc
oh lmfao automod deleted that spam message
magic coyote
distance from center of mass to orbit
it's actually the gravitational center, but that usually doesn't make a difference in such problems
quiet zinc
why the small g
quartz wind
bro these mistypes are not that deep
quiet zinc
yeah even if thats G its wrong
dimentionally too
just come back to basics
no muging up
i have a satalite on earth surface , earth mass is M , radius is R
whats the energy of the satalite which is still on the earth surface
quartz wind
am i supposed to set it as centripetal force
quiet zinc
you can do that to derive orbital velocity too yeah
just equate things properly
try
quartz wind
why is it not v = sqrt(GM/R)
quartz wind
yea ik i typed the wrong thing 😭
quiet zinc
yeah thats the orbital velocity , can you try to define the orbital velocity again
(its just so you dont forget it )
magic coyote
why is it not v = sqrt(GM/R)
Howd u find this
magic coyote
v = sqrt(GM/R)
thats the formula not the definition
quartz wind
i set Universal gravitation to centriptal force
Just tell me what you guys got and give me the process
Why can't i just add up TE of initial and subtract it by TE of final
magic coyote
which is what i was trying to get you to do
but you didnt know velocities
and hence this
quartz wind
bro 💀
quiet zinc
Why can't i just add up TE of initial and subtract it by TE of final
show us your attempt of doing this
you said everything right about the initial final kinetic potential energy, (you had the wrong idea of the velocity but thats corrected now i hope)
quartz wind
I got -GMm/2R for dissapated energy
magic coyote
show work
quiet zinc
I got -GMm/2R for dissapated energy
i mean yeah thats right
is there any doubt you had regarding this?
quiet zinc
ah freindly advise, dont skip over concepts man , you are doing great , just try to understand what things mean
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vital crag
Really not enough information to help
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brisk moth
i swear im not stupid but this question is throwing too much information at me
worthy wren
i swear im not stupid but this question is throwing too much information at me
add up the amount of time spent in the shower for each girl
this should give u the amount of time elapsed from 7 o clock to the last girl finishing her shower
runic needle
i swear im not stupid but this question is throwing too much information at me
do you understand the question
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@brisk moth Has your question been resolved?
old nacelle
a scienst has 2 radioactive substances each decaying at an exponential rate (not neccessealy 1/2 ). at 1 pm, he notes that the mass of first substance is 3 times that of the second. Later that day at 3 pm, he notes that the second substance now has 4 times the mass of the first. To the nearest minute, at what time did the two substances have equal masses
final saddleBOT
a scienst has 2 radioactive substances each decaying at an exponential rate (not...
old nacelle
no calculator