2 mistakes
#help-0
queen river
or there are more but im nearly blind
first, where is the negative sign?
second, you sub u to be cosx but you write it as sinx
Oh bruh you sub it as -sinx and it is still wrong for the positive and negative sings
signs
ocean sealBOT
queen river
👍
lone heartBOT
@lilac jolt Has your question been resolved?
queen river
stalin
safe tartan
lone heartBOT
safe tartan
i did a, b and c
but cannot do d
ill provide an image on sec
im not sure how i would deduce that O.o
queen cove
Well what happends when you add those vectors
w + y gets you back to R, so its overall 0
x + z get s you back to R, so its overall 0
So the overall sum is 0, as applying all 4 vectors gets you back to where you started
safe tartan
ah makes sense
i thought i had to use some algebra or something to show it adds to 0 but it was much simpler
thanks!
queen cove
np
lilac jolt
stalin
Sorry I didn't mean to stall, I was still studying and wasn't sure if Im gonna have another question and just didn't want to create another channel every question I had
safe tartan
.close
ornate gate
I've been asket to express this identities as rotation matrices and i don't even know were to start.... sin(α + β) = sin(α) cos(β) + cos(α) sin(β)
cos(α − β) = cos(α) cos(β) + sin(α) sin(β)
ornate gate
first I created two rotation matrices one for alpha and another one for beta. Multiplied them, but now i don't know how to go further
vale crag
yeah alright
multiplying the two rotation matrices for alpha and beta,
what rotation matrix should that result give you ?
@ornate gate
ornate gate
I multiplied rotation matrix A by rotation matrix B
ornate gate
i thought i'd create a transformation matrix, then just will need to multiply this AB matrix for alpha and beta
but the resuld didn't seem right
vale crag
there's something you should expect the result to be
like
if i rotate a vector twice
with angles alpha and beta
how much have I rotated my vector in total ?
ornate gate
alpha+beta angle (?)
vale crag
alpha+beta angle (?)
from this thing we just thought of,
if we multiply the two matrices for rotation for alpha and beta,
can we say something about the matrix which is the result of that multiplication ?
ornate gate
i'm doing the same job twice (?) the solution is just AB no need to mulitply it again (!?)
vale crag
i'm doing the same job twice (?) the solution is just AB no need to mulitply it ...
you seem kinda lost
i'm trying to make you see why we should find sin(a+b) and cos(a+b) in the result of the multiplication
maybe we'll just do it a bit more step by step
and check what you did
so, what's the rotation matrix for angle alpha ?
ornate gate
[cos(a)-sin(a)],[sin(a),cos(a] (sorry, no alpha symbol)
[cos(b)-sin(b)],[sin(b), cos(b)] for beta
ornate gate
ok, sorry
vale crag
[cos(a)-sin(a)],[sin(a),cos(a] (sorry, no alpha symbol)
[cos(a)-sin(a)], it seems like there's only one number on this line
you meant [cos(a), -sin(a)]
ornate gate
[-sin(a)sin(b)+cos(a)cos(b), -sin(a)cos(b)-sin(b)cos(a)],[cos(a)sin(a)+sin(a)cos(b), cos(a)cos(b)-sin(b)sin(a)]
vale crag
yeah alright
now i just want to know if you're in high school or university (or something else idk) ?
ornate gate
first semester of university...
vale crag
alright
did they just throw the matrix multiplication computation in your face ?
or did they explain why it is like this ?
ornate gate
I learned it in a mechanical way, i don't really understand what's going on actually
vale crag
yeah alright, I should have asked that first before questioning you intensively
lone heartBOT
@ornate gate Has your question been resolved?
ornate gate
No
vale crag
the nice idea behind matrix multiplication, is that it represents applying successive transformations on some vector
i.e. if I apply AB to some vector v
it's the same as applying B then applying A to v (note the order, it's like function composition)
so the thing I was trying to make you think about at the beginning
is that if you multiply the rotation matrix for alpha and the rotation matrix for beta
then the result is the rotation matrix for (alpha+beta)
@ornate gate
lone heartBOT
@ornate gate Has your question been resolved?
copper ginkgo
How do I find the max and min value of 4x+y^2 under the condition that 2x^2+y^2=4?
copper ginkgo
I already found the max value using a derivative. But I don't know how to find the minimum
alpine sable
Lagrange multipliers?
copper ginkgo
what's that
flat ore
I don't think there is a minimum
alpine sable
[
\grad \map f{x,y} = \lambda\grad\map g{x,y}
]
ocean sealBOT
flat ore
since y^2=4 - 2x^2, you have to find the extrema for 4x+4-2x^2
copper ginkgo
err not sure if this is how I'm supposed to solve it
copper ginkgo
There is
flat ore
flat ore
ah yes
flat ore
$2x^2 = 4 - y^2$
$x^2 = 2 - (y^2)/2$
$x = \pm \sqrt{2 - \frac{y^2}{2}}$
ocean sealBOT
Swastik
ancient saddle
The answer sheet says it should be -4sqrt(2)
Consider that the restriction is an ellipse, what is the interval of possible values for x?
Then remember how you find minimum and maximum given an interval
flat ore
hmm I think geometrically finding that is easier
copper ginkgo
Consider that the restriction is an ellipse, what is the interval of possible va...
hmm not sure I remember how
ancient saddle
hmm not sure I remember how
Do you remember the ellipse equation?
copper ginkgo
not sure
ancient saddle
Ok then I think Swastik was doing something that didn't need that 🤔
I don't know how to do it without the ellipse equation
You could review that if there's no other way
copper ginkgo
$2x^2 = 4 - y^2$
$x^2 = 2 - (y^2)/2$
$x = \pm \sqrt{2 - \frac{y^2}{2}}$
I see
flat ore
**Swastik**
see 2-y^2/2 has to be greater than 0 for it to be defined
and y^2 is always postive
so if we maximize 2-y^2/2 we get 2 (when y =0) and min is 0
i.e. x lies between (-sqrt2, sqrt2)
now you can plug these values of x into that quadratic and see which is the lower
that should be the minimum
copper ginkgo
I give up lol
lone heartBOT
lone heartBOT
Available help channel!
alpine sable
help
limpid spade
hi
alpine sable
its using Cos
undone lantern
,rotate
ocean sealBOT
undone lantern
use the sin rule
alpine sable
i hjave to use cos
undone lantern
fine use
$cosA = \frac{b^2 + c^2 - a^2}{2bc}$
ocean sealBOT
roxyit
chilly nova
Help
undone lantern
do you remember trignometry?
undone lantern
welp go and revise your notes then
chilly nova
welp go and revise your notes then
if ur not helping then whyd u even bother to comment
🙄
undone lantern
you need to at least be familiar with the basics of what you're trying to solve
no point in solving questions without learning the theory first
chilly nova
you need to at least be familiar with the basics of what you're trying to solve
Its not for me, rather a friend
They just asked me to post this
Since im already here
undone lantern
being used by friend, how nice
anyway you need to use the definition for sin
to solve that
chilly nova
still not helping
lone heartBOT
@chilly nova Has your question been resolved?
tardy stag
go tell your friend to revise their notes then
queen socket
im so confused on what i need to do next
lone heartBOT
im so confused on what i need to do next
radiant oasis
I think the first step would be to multiply both sides by x
radiant oasis
x tan38= 10
radiant oasis
Because you are multiplying both sides by x
fallow sluice
radiant oasis
Then you would do inverse tan
fallow sluice
Are u familiar with this
radiant oasis
Thats for law of sines right?
fallow sluice
Yh in a Triangle
tardy stag
do not bring the law of sines into this, we don't need it for right triangles
radiant oasis
I don't think you need law of ainea for this since its a right triangle
shy dove
you don't need law of sines. They were on the right track
radiant oasis
Yeah law of sinea is not needed sorry for the confusion @queen socket
queen socket
i inverse it
radiant oasis
Mhm
queen socket
um
radiant oasis
And then you would get x=...?
queen socket
wait im slow
radiant oasis
Ok no problems
queen socket
wait wtf
queen socket
oh
radiant oasis
Go back to x*tan38=10
radiant oasis
Since x is being multiplied by tan38, you'd divide
radiant oasis
No
fallow sluice
Tan38 is a number
shy dove
Think of tan(38) as a value
radiant oasis
And you'd divide both sides by tan38 to isolate the x
radiant oasis
Here let me check
shy dove
,w tan38 = 10/x
ocean sealBOT
shy dove
fun
radiant oasis
Yup you got it es
shy dove
thought it'd give approximated value
queen socket
TYSM
radiant oasis
No problem! Out of curious are you studying for finals?
queen socket
nope
radiant oasis
Oooh i have that next year
queen socket
i had the worst math teachers for the past 3 yhears so im rlly behind
radiant oasis
Here's a triangle, we know ita a triangle, now PROVE IT
queen socket
ONG BRO
radiant oasis
its awful
radiant oasis
ik
alright well should probably close if you have no more questions bc were holding up the room, but are you all good?
lavish monolith
Find the clamped cubic spline S(x) that passes through data points given in the table
X=-1, 0 ,1
Y=1 ,2 ,-1
with the derivative boundary conditions S′(−1) = 1 and S′(1) = 1
lavish monolith
no matter what i do i can't find it
no
we didn't use this way
i tried this way i end up with tons of linear equation and i don't know to simplify it 😦
limpid turret
Bro how'd you write all this in one minute?
lone heartBOT
@lavish monolith Has your question been resolved?
lavish monolith
Bro how'd you write all this in one minute?
chat gpt probably
lavish drift
why did you guys ban @alpine sable#6950
limpid turret
why did you guys ban <@456226577798135808>#6950
For using chatgpt, which is considered spam since it does not do actual math.
lone heartBOT
@lavish monolith Has your question been resolved?
tacit arch
why did you guys ban <@456226577798135808>#6950
Ask mods, not spamming people's help channels
void solar
interesting question, but why
silver portal
its 10% of my final grade according to my teacher
void solar
so what do you need help with
use whatever equations you've learnt in class and try and outline it
you can put a picture on desmos
silver portal
I’ve already tried that
It’s impossible
All the videos ive seen on YouTube are examples of logos
Not pictures like those
wary stream
It's not impossible
void solar
well obviously you cant just copy a video
wary stream
You're going to create multiple equations with restrictions
void solar
do equations you've learnt about like circles or ellipses or quadratics or linear lines
you could even do it bit by bit with just linear lines if you create restrictions.
but you probably wont get full marks
wary stream
With lines, it's going to take forever, adding quadratics, it'll be easier
silver portal
how do i get that in desmos tho lol
silver portal
Its easy to draw it but it aint easy to put that on desmos
like I’ve never done it before thats why it look’s impossible
wary stream
You don't need this one, I don't think, OP was just asking about the outline of the image so the inner stuff might not be needed
wary stream
how do i get that in desmos tho lol
Do you know the general equation of a quadratic?
void solar
idk if this would help but i have an example i made a while ago
https://www.desmos.com/calculator/9aprketqrn
wary stream
Because in desmos, you can add sliders
silver portal
whats a slider
void solar
sliders would be a smart way to do it
wary stream
whats a slider
A thing that slides
Type in ax^2 + bx + c in desmos
That's the general form of a quadratic
The site then will say something about a slider for a, b, and c
So then you can move that around to pick a number for a, b, and c
silver portal
So like this
silver portal
How can i use that same line to finish the rest of the head
wary stream
You might not be able to, you might have to create multiple lines
void solar
you slowly trace it bit by bit by using multiple lines
lone heartBOT
@silver portal Has your question been resolved?
austere compass
I stumbled upon this and I have no idea how to start lol
The linear map $F \in L(\mathbb{C}{\leq 3} [t], \mathbb{C}{\leq 3}[t])$ is given by $F\left(\sum^3_{i=0} \alpha_i t^i\right) = \sum^3_{i=0} \beta_i t^i$, where
$$
\beta_0 = 3\alpha_0 - \alpha_1, \quad \beta_1 = \alpha_0 + \alpha_1, \quad \beta_2 = 3\alpha_0 + 5\alpha_2 - 3\alpha_3, \quad \beta_3 = 4\alpha_0 - \alpha_1 +3\alpha_2 - \alpha_3.
$$
Determine the Jordan normal form of $F$.
ocean sealBOT
Levens
alpine sable
do you know how to get the matrix associated with F?
alpine sable
what is the canonical basis for your vector space?
austere compass
mmmm if it were R_{<=3}[t] i'd say {1,t,t^2,t^3}... for complex polynomials im not so sure
but ig the same?
alpine sable
yes, thats correct, its the same
austere compass
ok nice
austere compass
okay.. would we look at the coefficients for alpha or for beta? like e.g. for $\beta_1$ is it $(\alpha_0 + \alpha_1)t$ or is it $\alpha_0 + \alpha_1 t$?
ocean sealBOT
Levens
austere compass
okay, lets see how far i get thanks
is it
$$
\left[\begin{array}{cccc}
3 & -1 & 0 & 0 \
1 & 1 & 0 & 0 \
3 & 0 & 5 & -3\
4 & -1 & 0 & 0
\end{array}\right]?
$$
ocean sealBOT
Levens
alpine sable
yup, correct
austere compass
yay
lone heartBOT
@austere compass Has your question been resolved?
austere compass
So i got the eigenvalues 0,2,5 and 2 has an alg. mult. of 2 so how do I find out to sort this whole thing? so ik that theyre on the diagonal and EV 2 is a 2x2 block somewhere..
alpine sable
find the evecs of 2
lone heartBOT
@austere compass Has your question been resolved?
austere compass
find the evecs of 2
why
alpine sable
I think thats how the method for jordan form works
find the geo mult
then use the evecs to fill in the block
austere compass
ahhh i see for the geo mult
i only got 1 vector for 2
same for the other two evals
alpine sable
yeah
the other ones have to have geo mult 1
so for ev = 2, do step 3 here: https://math.stackexchange.com/questions/1570526/computing-the-jordan-form-of-a-matrix
austere compass
so for ev = 2, do step 3 here: <https://math.stackexchange.com/questions/1570526...
step 3? but they already formed their JNF there
alpine sable
hm yeah, ig you dont need the P matrix
ig you are finished then after finding the geo mults
austere compass
that's weird, i only got one evec for 2 for some reason, but on wolfram and everywhere else the jordan block for 2 is 2x2...
alpine sable
,w jordan [[3,-1,0,0],[1,1,0,0],[3,0,5,-3],[4,-1,0,0]]
ocean sealBOT
alpine sable
looks fine to me
alpine sable
that's weird, i only got one evec for 2 for some reason, but on wolfram and ever...
what are you getting?
yes, thats what we should get
2 has alg mult 2, geo mult 1
so it appears twice on diag
and theres a 1 on the off diag
austere compass
and theres a 1 on the off diag
i thought that meant that its geo mult is 2
so when would a 0 appear above the 2s?
hypothetically
austere compass
and does the order matter? like could i just write
$$
\left[\begin{array}{cccc}
2 & 1 & 0 & 0 \
0 & 2 & 0 & 0 \
0 & 0 & 5 & 0\
0 & 0 & 0 & 0
\end{array}\right]?
$$
ocean sealBOT
Levens
austere compass
ah i see, i just looked it up, apparently it doesnt matter (i think)
but abt that 1 above the 2s... why is it there?
alpine sable
theres one block because geo mult = 1
if geo mult was =2, then theres 2 blocks (ie no 1 on the off diag)
austere compass
ohhh ok, so the geo mult tells us how many blocks there are for each eval
alpine sable
yes
austere compass
bet thanks
i have another question, should i open a new channel or should i jsut say it here
alpine sable
a new channel would be a good idea so that the new question gets pinned
austere compass
Let $\mathbb{F}$ be a field and $n \in \mathbb{N}$. Let $A \in GL(n,\mathbb{F})$ be triangulable (similar to an upper triangular matrix).\
\
i) Show that $A^{-1}$ is also triangulable.\
\
ii) Determine a Jordan normal form of $A^{-1}$.
austere compass
So for i) i found something on mathstacks:
But what about ii)?
ocean sealBOT
Levens
lone heartBOT
@austere compass Has your question been resolved?
fair cairn
I'm trying to calculate the expected number of tosses of a fair coin, such that we get 3 consecutive heads
fair cairn
This is the method I have used, and it gives the wrong answer, but I don't see why
My final answer is 10, which is based on calculating the expectations of X and 6, where X is the indicator variable, if a sequences of 3 heads starts at a given index, and Y is the sum of X over the entire sequence
I based it off my teachers explanation of infinite money theorem, so I can provide a similar example of my method being used, maybe someone can point me to why it's not applicable potentially? Or a silly mistake
lone heartBOT
@fair cairn Has your question been resolved?
lone heartBOT
@fair cairn Has your question been resolved?
desert prism
dont understand where answer is coming from
lone heartBOT
dont understand where answer is coming from
desert prism
wouldnt i just use the distance formula d=|Ax+by+cz+d|/sqrx^2+b^2+c^
plug in the given values then isolate for b?
gleaming ridge
Correct
desert prism
i guess im doing it wrong as im getting incorrect answers
gleaming ridge
Show me
desert prism
ooooo
one minute
oh hold up
cant rlly show you cause im on pc n im solving on paper
but eventually i get to 8=24+12b/sqr(65 + b^2)
multiple the denominator to the other side
desert prism
but eventually i get to 8=24+12b/sqr(65 + b^2)
actually..this might sound goofy but whats after this step😭
gleaming ridge
And then yes, you multiply by the denominator both sides
You end up with:
8 . sqrt(65+b²) = |24-12b|
desert prism
mhm
gleaming ridge
As you know, |x| is either x or (-x)
x when x>0
-x when x<0
So, we have two cases here
Solve 8 . sqrt(65+b²) = 24 - 12b
And 8 . sqrt(65+b²) = -24 + 12b
desert prism
oooooooo
so yk how its 8sqr...
can we deal with the sqrt without dealing with the 8
like could i initially just square both sides?
ooo id just have to square the 8 too
okok makes sense, thank you!
.close
sterile sparrow
double integral calc 3
lone heartBOT
double integral calc 3
sterile sparrow
i'm trying to figure out how to set up the integrals
since it's bounded by z=16-x^2 and y =5, would it be 0 <= y <= 5 and 0 <= x <= 16-x^2 ?
modern tundra
yes'
desert prism
nah bros doing math help and valorant at the same time😭(sorry for interrupting)
modern tundra
nah bros doing math help and valorant at the same time😭(sorry for interrupting)
what rank are you ?
sterile sparrow
.close
slender thistle
What is standard deviation explained to an 8th grader?
lone heartBOT
What is standard deviation explained to an 8th grader?
@slender thistle Has your question been resolved?
nocturne iris
how do i find the antiderivative of xcos(x²)?
lone heartBOT
how do i find the antiderivative of xcos(x²)?
solid sonnet
how do i find the antiderivative of xcos(x²)?
Hint: use u = x²
nocturne iris
Hint: use u = x²
∫ cos(u) du = sin(u) + C ?
slender quiver
yeah but what happened to the first x
nocturne iris
square root
solid sonnet
∫ cos(u) du = sin(u) + C ?
∫ cos(u) dx
Now make it in terms of du
nocturne iris
∫ cos(u) dx
Now make it in terms of du
it would be sin(x²)/2 then but i didn't know this formula coulb be applied to anything else different from x
solid sonnet
it would be sin(x²)/2 then but i didn't know this formula coulb be applied to an...
Yes exactly you got it
tardy stag
look up u substitution
solid sonnet
Good job
solid sonnet
yeah but what happened to the first x
u = x²
du = 2x dx
du/2x = dx
x * du/2x = 1/2
Oh my bad thought you were the OP asking the question
nocturne iris
.close
nocturne iris
hello again
lone heartBOT
hello again
nocturne iris
$\sin(x) \cdot e^{\ln(\cos(x))}$
digital magnet
What is exp nnotation
ocean sealBOT
lilisworld
digital magnet
ah
nocturne iris
what is the antiderivative?
wind cloak
can you simplify it first
digital magnet
^^
digital magnet
good
wind cloak
yes so now just integrate that
tulip cypress
I need help
lone heartBOT
I need help
tulip cypress
This question, How do i solve it?
past sundial
hint:
1 is 1/6
2 is 1/6
3 is 1/6
4 is 1/4
5 is 1/4
even numbers: 24
odd numbers: 135
tulip cypress
ok
past sundial
for one spin ^
tulip cypress
odd numbers probability for a spin is 49/144
so even numbers one spin is 144/144-49/144 =
95/144
am i corect
past sundial
i don't really get what you're saying
tulip cypress
oh i didnt calirfy
clarify*
since it was two spins, probability of getting an odd number on both spins is (7/12) squared
so that is 49/144
and so if one of those even, you subtract 49/144 from 1, and you get 95/144
probability for even
on one spin if it's spun 2 time
past sundial
i think it's 15/58 but i could be wrong
full relic
even numbers: 5/12
That's right.
upbeat hornet
How can I find a curve which all lines of the form $$ x + by + b^{\frac23} = 0 $$ are tangent to?
ocean sealBOT
(a + b)^2 = a^2 + b^2
tacit arch
Are tangent to what
upbeat hornet
The curve
tacit arch
There are infinitely many curves
Are you just working with polynomials, quadratics perhaps
upbeat hornet
I plotted some lines and couldn’t think of a curve whose tangents are all of this form
tacit arch
**(a + b)^2 = a^2 + b^2**
Find the tangent line for a general quadratic and set the tangent line to these coefficients
upbeat hornet
I’ll try that, thanks
tacit arch
**(a + b)^2 = a^2 + b^2**
Probably will help to pick a point on the curve. y=0 implies x=?
lone heartBOT
@upbeat hornet Has your question been resolved?
upbeat hornet
I tried plotting a few more lines
I don’t think it’s a parabola
After some guessing, I think it’s xy^2 = -c for some c
.close
lone heartBOT
lone heartBOT
Available help channel!
finite mesa
how to do surds on texit bot
lone heartBOT
how to do surds on texit bot
echo socket
$\sqrt{x}$ and $\sqrt[n]{x}$, but ask such questions in #latex-help next time
ocean sealBOT
A Lonely Bean
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
echo socket
proven leaf
it doesn't like hashtags
echo socket
Oh
proven leaf
$#$
ocean sealBOT
XxMrFancyu2xX
finite mesa
.close
marsh silo
In a sine/cos graph how do I figure how the second intercept. I.e f(x) = -2cos(3x)-1
in this example i already know that the reference angle will be pi/3
the problem I have is when finding the second point
Which is when I draw it on the unit circle I know that it should be in the 1st and 4th quadrant, so do I need to 2pi-pi/3 or 3pi/4 + pi/3 to find it?
marsh silo
<@&286206848099549185>
tardy stag
well it definitely wouldn't be 3pi/4 + pi/3, that's in the third quadrant
for the unit circle, when you see cosine think horizontal, and when you see sine think vertical
you want the angle that is at the same horizontal point (x-value) as your pi/3
marsh silo
so basically basically I reflect the triangle formed?
tardy stag
yeah
tardy stag
generally it's best to keep the hypoteneuse as 1
but pi/3 is the angle not the slope. It's an angle of pi/3 radians
marsh silo
okay
alpine sable
you have the reference angle and you want cos to be negative you'd want to consider the equivalent angles in the second and third quadrants
alpine sable
he wants the other angle where cos is positive
unless his question changed, i thought he's working on finding the x intercept of -2cos(3x)-1
marsh silo
<:nozoomi:651816986837254148>
you are correct
I also just realised that I wrote the questio incorrectly it should be f(x) = -2cos(3x)+1
alpine sable
I also just realised that I wrote the questio incorrectly it should be f(x) = -2...
ah cool
you're working with the first and fourth quadrant then
marsh silo
yep
so first value would be pi/3
then to find the second value i would need to do 2pi- pi/3 since the reference angle is always from a value of pi right?
alpine sable
yes
tardy stag
for cosine that works, for sine it won't
but yes for cosine it will work
marsh silo
yes for sine it is just inverse right?
alpine sable
it's counterclockwise counting upwards of positive multiples of n
otherwise negative multiples of n
tardy stag
$sin(-x) = -sin(x)$ yes
ocean sealBOT
kitten.in.a.teacup
tardy stag
green is cosine, orange is sine
marsh silo
okay
so let's say the equation is now f(x) = -2sin(3x)+1
the first intercept is pi/6
alpine sable
You didn't finish the prevoius question though? Lol
If you're working on [0, 2pi] there's more than the two you pointed out btw
alpine sable
the first intercept is pi/6
yes ofc that part should be trivial
alpine sable
one period = 2pi/3
what
marsh silo
yes ofc that part should be trivial
and then the second point is just the inverse so -pi/6
alpine sable
no yeah that's correct, one period of f(x) is 2pi/3 long
i know the period of f(x) is 2pi/3 lol
i thought he meant the period of a generic periodic function
alpine sable
and then the second point is just the inverse so -pi/6
depends on the interval of concern
[0, 2pi/3] wouldn't include -pi/6, would it?
marsh silo
I would just adjust for domain
alpine sable
sure
marsh silo
also one more thing
if the reference angle is negative
i.e. cos (x) = -√3/2
which becomes -pi/6
is the reference angle just pi/6 or -pi/6
so to find the second point do I do pi +( - pi/6) or pi - (-pi/6)?
alpine sable
is the reference angle just pi/6 or -pi/6
ocean sealBOT
alpine sable
is that -sqrt(3)/2?
ocean sealBOT
marsh silo
is that -sqrt(3)/2?
ye
alpine sable
ye
it's not? cos(pi/6) = sqrt(3)/2
you want -(sqrt(3))/2
if you're talking about the reference angle as to what you should work with to get 5pi/6 then okay fair pi/6 is right
viscid lily
Hrllo
viscid lily
The first one please
granite inlet
write out what it means, as it relates to the polynomial's factors, for $\csc{\alpha}$ and $\cot{\alpha}$ to be roots
ocean sealBOT
Blue Guilmon
viscid lily
Can you elaborate I'm new to this topic
granite inlet
if $y$ is a root of a polynomial, the factor theorem tells us that $(x-y)$ must be a factor of the polynomial
ocean sealBOT
Blue Guilmon
viscid lily
Ok
granite inlet
so if $\csc{\alpha}$ is a root, then the polynomial has a factor which is $(x-\csc{\alpha})$
ocean sealBOT
Blue Guilmon
viscid lily
So you mean to find the equation by using roots and then plugin and try the values for all?
granite inlet
what's the degree of the polynomial in question
viscid lily
2
granite inlet
so how many roots can it have maximum
viscid lily
2
granite inlet
you were given both roots, and so you know what the polynomial factors into completely
wet nest
cosec^2 x - cot^2 x =1
viscid lily
Hmm
granite inlet
this channel is occupied
wet nest
(Cosecx-cotx)(cosecx+cotx)=1
(-b/a)(rootd/a)=1
Now square and simplify
granite inlet
$(x-\csc{\alpha})(x-\cot{\alpha})=ax^2+bx+c$ you know this because this is how the polynomial in question factored for the coefficients $a$ $b$ and $c$. Expand out the left and you'll get 3 equatiosn for $a$ $b$ and $c$, for example, for $a$ you should get $1=a$ by setting the coeffcients equal to each other. For $c$ you should get $\csc{\alpha}\cot{\alpha}=c$ what about to $b$? Plug these values for $a$ $b$ and $c$ to evaluate the expressions in each of the multiple choices to see which is true using what you know from trig.
wet nest
b^2(b^2-4ac)=a^4
b^4-4acb^2=a^4
a^4-b^4+4acb^2=0
ocean sealBOT
Blue Guilmon
wet nest
@viscid lily
viscid lily
Ok
wet nest
Basic trig identity
viscid lily
I am trying with x²-sor+por=0
wet nest
sin^2x+ cos^2x=1
Divide by sin^2x
1+cot^2 x= cosec^2x
viscid lily
Ok
granite inlet
(Cosecx-cotx)(cosecx+cotx)=1
(-b/a)(rootd/a)=1
Now square and simplify
better question is how did you get the second equation here from the first step
wet nest
Basic quadratic identity
granite inlet
root d/a?
wet nest
Sor = -b/a
|Difference of roots|= sqrt(b^2-4ac)/a
granite inlet
discriminant?
wet nest
Yeah
viscid lily
Yes
granite inlet
ah ok yeah both methods work then
viscid lily
Got it thanks btw
wet nest
Solve with this, you will get only 3-4 mins to solve this in exam
viscid lily
Hmm
granite inlet
this method would only work for quadratics
because of the quadratic formula
unless we were given an explicit formula for factoring higher degree polynomials in question
viscid lily
Yes
viscid lily
But I want to explore mathematics
granite inlet
then in general the way I told you is how you would approach the problem, but they both work
if you have a formula relating the roots to the coeffcients it cuts the work down a lot
wet nest
There is a generalized sequence for sum of roots for any degree
wet nest
What memorising?
granite inlet
well either way a problem like this
viscid lily
but memorizing that is more work than solving the system of equations
Competitive approach to solve problems needs to have less effort I think
But at last the path to the question matters
granite inlet
unless you have a trick or formula relating coefficients to roots like the quadratic in this case
viscid lily
What you learn from it
wet nest
ax^3+bx^2+cx+d
Sum of roots = -b/a
Sum of roots taken two at a time = c/a
Product of roots = -d/a
Just alternate minus sign
granite inlet
degree 4?
actually tbh
if it is generalized for any number of roots what's degree n
wet nest
ax^4+bx^3+cx^2+dx+e
Sum of roots = -b/a
Sum of roots taken two at time = c/a
Sum of roots raken three at time = -d/a
Product of roots = e/a
granite inlet
so
$\frac{-a_{n-1}}{a_n}$ is always the sum of the roots
everything in between is the sum of roots taken 2 to $n-1$ at a time
and then product of the roots is always $(-1)^n\frac{a_0}{a_n}$
ocean sealBOT
Blue Guilmon
granite inlet
alternating signs in between
wet nest
Yeah
wet nest
I don't know where this is derived from
wet nest
let x1,x2,x3,x4 be 4 roots
Then sum of roots taken two at time = x1x2+x1x3+x1x4+x2x3+x2x4+x3x4
granite inlet
ah all possible combinations of 2
wet nest
Yeah
granite inlet
I'm not sure how useful that would be beyond something like degree 4
because that gets pretty huge
wet nest
Only sum and products are necessary
granite inlet
I suppose but in this case our roots were arbitrary trig functions
you're talking about sums and products of potentially 4 different trig functions
or more
idk I guess it's a neat tool to use but I don't see either method as being particularly better
because at the end of the day expanding the poly is also just doing multiplication/addition and you just get sums and products equal to the coeffcients
tbh maybe that's how you prove this sequence
I might look into that later
anywho hope we answered your question!
wet nest
idk I guess it's a neat tool to use but I don't see either method as being parti...
For jee, the method with less calculation is best
There is a limited amount of time
granite inlet
no clue what jee is but I'm also not an engineer I'm assuming from the title of this that's what it is for
wet nest
Jee is a competitive exam with 1 million candidates each year and only 5000 selection seats
lone heartBOT
@viscid lily Has your question been resolved?
viscid lily
Yes
alpine sable
I am having trouble on how to solve this problem.
lone heartBOT
I am having trouble on how to solve this problem.
fading moth
do you know how to convert to polar coordinates?
alpine sable
I believe so
fading moth
what's the conversion?
remember, your goal is to convert the r and thetas into x and y
lone heartBOT
Available help channel!
glossy hill
hi
lone heartBOT
lone heartBOT
✅
vapid shuttle
.close
lone heartBOT
earnest tulip
f(x) = x * (cos(π)^2 + sin(π)^2) - log(e^x) + √(x^2) + (e^(iπ) + 1) 0 + (e^(ln(x)) - x) * (atan(tan(45°)) / π)
lone heartBOT
Available help channel!
dire gale
The five-number summary for all students' scores on an exam is 32, 42, 67, 67, 99. Suppose 120 students took the test, the mode value is mostly ...
A. 67
B. Cannot be determined.
C. 99
D. 32
dire gale
I'm a bit confused by this.
The answer is 67, but I'm not sure how
I tried to find it with Q3-Q2 < Q2-Q1. Which means it's negatively skewed so Mode > Median > Mean
steep harbor
Um
And 120 divided by 5 is 20
Wait no
That's kind of irrelevant
But anyways 67 appears the most
steep harbor
Where is Q2 and Q3?
vale wigeon
And 120 divided by 5 is 20
120/5 is not 20
dire gale
32 = minimum value, 42 = Q1, 67 = Q2 , 67 = Q3, 99 = Maximum
steep harbor
120/5 is not 20
My bad
vale wigeon
32 = minimum value, 42 = Q1, 67 = Q2 , 67 = Q3, 99 = Maximum
all the values between Q2 and Q3 are 67 aren't they
dire gale
all the values between Q2 and Q3 are 67 aren't they
I'm not sure
vale wigeon
why aren't you sure?
about a quarter of your data points fall between Q2 and Q3.
what could those data points be equal to if not 67?
dire gale
Ah I get it, you could also say 25% of the data is 67 right?
vale wigeon
give or take, yes
alpine sable
(1/2)² = 1/4 ----- (i) but (2/4)² = 4/8 = 1/2 ----- (ii) even though in eq (i) 1/2 equals to 2/4 the value of squares come different
lone heartBOT
(1/2)² = 1/4 ----- (i) but (2/4)² = 4/8 = 1/2 ----- (ii) even though in eq (i) 1...
vapid shuttle
4^2 does not equal 8
the value of the squares are not different
you just squared the second one wrong
but don't occupy multiple channels
.close
lone heartBOT
vapid shuttle
the channel will close soon
alpine sable
btw which grade you are
vapid shuttle
why?
alpine sable
just like that
vapid shuttle
12th
alpine sable
i am in 10th
for some second i thought i broke mathematics but i was wrong
that's why mathematics is great
vapid shuttle
couldn't agree more
rose sigil
lol
lone heartBOT
Available help channel!
median oar
when i look at something like this (summary in R for simple linear regression)
what's the X for
median oar
im guessing (intercept, estimate) is the point estimate of alpha
im also guessing that std. error is the standard error of the alpha estimate
t value is comparing it to alpha = 0
and probability is just the probability of that being true
but idk what X's row is
lone heartBOT
@median oar Has your question been resolved?
vale crag
but idk what X's row is
lm is linear regression right ?
vale crag
it's prolly just for the coefficient in front of X
median oar
OH
vale crag
Y = aX+b
median oar
that makes so much sense
vale crag
the X line is prolly the estimate for a
severe abyss
hi help pls
lone heartBOT
hi help pls
echo socket
!status
lone heartBOT
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'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
You know about sine and cosine law??
lone heartBOT
@severe abyss Has your question been resolved?
severe abyss
why are you finding cos(A) when you were asked for angle C though?
sorry my fault
i realized that now
mellow grail
hi help pls
!show
lone heartBOT
hi help pls
Show your work, and if possible, explain where you are stuck.
Available help channel!
scenic wing
vale wigeon
you deleted your original message and now the channel will close abruptly. claim a new one.
lone heartBOT
Available help channel!
austere compass
Let A be in GL(n,F) and let A be triangularisable. How would one determine a jordan normal form of A^(-1)?
lone heartBOT
Let A be in GL(n,F) and let A be triangularisable. How would one determine a jor...
@austere compass Has your question been resolved?
austere compass
<@&286206848099549185>
austere compass
Would
$A^{-1}$:
$$
A^{-1} = S^{-1}J^{-1}S = S^{-1} \begin{bmatrix} J_{r_1}(\lambda_1)^{-1} & & \ & \ddots & \ & & J_{r_k}(\lambda_k)^{-1} \end{bmatrix} S
$$
with
$$
{J_A}_r(\lambda)^{-1} = \begin{bmatrix} \frac{1}{\lambda} & -\frac{1}{\lambda^2} & \frac{1}{\lambda^3} & \ldots & (-1)^{r-1}\frac{1}{\lambda^{r-1}} \ 0 & \frac{1}{\lambda} & -\frac{1}{\lambda^2} & \ldots & (-1)^{r-2}\frac{1}{\lambda^{r-2}} \ 0 & 0 & \frac{1}{\lambda} & \ldots & (-1)^{r-3}\frac{1}{\lambda^{r-3}} \ \vdots & \vdots & \vdots & \ddots & \vdots \ 0 & 0 & 0 & \ldots & \frac{1}{\lambda} \end{bmatrix}
$$
be a good fit?
ocean sealBOT
Levens
austere compass
the lambdas are the evals btw
lone heartBOT
@austere compass Has your question been resolved?
rapid plover
hi
rapid plover
1/2sin(x-pi/2)+3
how do i find x intercepts
narrow rain
desmos
unless you can't use that
rapid plover
cant
narrow rain
It's just the points that intercept the x axis
rapid plover
we are not allowed
narrow rain
I could probably answer this if I like really focused on figuring it out but idk off the top of my head
upbeat hornet
1/2sin(x-pi/2)+3
how do i find x intercepts
set it equal to 0 and solve
rapid plover
yes i tried
upbeat hornet
what got you stuck
rapid plover
but im not getting
well first trasnpose 3
then 1/2
sin(x-pi/2)=-6
cant go further
queen river
sin(x-pi/2)=-sin(pi/2-x)=-cosx
queen river
just calculate it
queen river
cosx=6
queen river
So x does not belongs to real numbers
queen river
are you sure
rapid plover
yes
gray isle
did you type the correct function?
rapid plover
queen river
there wont be x intercept
rapid plover
queen river
Bro
rapid plover
there will be imaginary line
gray isle
as this doesn't have any x-intercepts
rapid plover
ok take the line as 3.5
thank you all for the help. its 10:20 pm and iam tired now
i hope you all have a good day/night ahead
.close
stiff surge
lone heartBOT
stiff surge
I am not sure how to draw the graph
So like for x intercept I got 1 and -4
and y intercept I got -4
For minimum point or the turning point I got (-3/2,-25/4)
lone heartBOT
@stiff surge Has your question been resolved?
stiff surge
.close
fair cairn
An elevator operates in a building with 10 floors. One day, n people get into the
elevator on the ground floor (Floor 0), and each of them chooses to go to a floor
selected uniformly at random from 1 to 10. What is the expected number of floors in
which exactly one person gets out? Justify your answer.
fair cairn
No idea how to approach this question, I tried infinite monkey theorem, or a variation of it
Should I be thinking in terms of conditional expectation?
vale wigeon
@glossy echo please open your own channel. #❓how-to-get-help has the instructions.
also phrase your question more clearly there.
glossy echo
ok
vale wigeon
An elevator operates in a building with 10 floors. One day, n people get into th...
find the probability of floor 1 getting exactly one person walking out on it
multiply this probability by 10 and you have your expectation
fair cairn
multiply this probability by 10 and you have your expectation
surely you can't just multiply it by 10, because its dependent on the number of people, which decreases as the lift goes up. Moreover, the chances of each person leaving increases as the lift goes up, as everyone has to exit the elevator by floor 10, so it seems to be all very dependent on each other and circular?
vale wigeon
nah you can
let X_i be the random variable that's 1 if floor i has exactly one person walking out and 0 otherwise
i believe it is clearish that all the X_i are identically distributed, even if not independent
X := sum[1,10] X_i is the number of floors with exactly one person walking out
E[X] = E[sum[1,10] X_i] = 10 E[X_1] is what i am saying
fair cairn
how would you calculate the probability of exactly one person leaving?
vale wigeon
the probability they choose floor 1, times the probability everybody else chooses a floor other than 1, times how many people there are
basically like using a binomial distribution
the number of people leaving on floor 1 follows a binomial distribution with number of trials n and success probability 1/10
fair cairn
so at floor 1 we would have 1/10 * n * (9/10), but number n varies by floor?
lone heartBOT
@fair cairn Has your question been resolved?
lone heartBOT
@fair cairn Has your question been resolved?