help-23

[potent seal]

You can find a-b from there

[gentle edge]

How

[potent seal]

(a+b)²-4ab= (a-b)²

[gentle edge]

O

Ok

[potent seal]

Ye

And then proceed

[gentle edge]

Then u got w equations

[potent seal]

Ye

Once you get the roots you can simplify the quadratic

[gentle edge]

Yep

Got it

.close

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[royal spruce]

Anyone have any stats knowledge?

[royal spruce]

I would be willing to chat as well.

[safe radish]

No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/

[old chasm]

what do you really want to achieve?

[safe radish]

@royal spruce Has your question been resolved?

[safe radish]

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[sterile glacier]

guys am i wrong

[lean otter]

Can't se properly What's your third row operation?

[sterile glacier]

which one u mean

[lean otter]

Final

Which gave you the upper. Matrix

[sterile glacier]

R3: 3R2 - R3

I take R2 times by 3 and minus R3 , put it into R3

[lean otter]

Check again, I think that wouldn't work.

[sterile glacier]

Wait, how really gaussian elimination works?

is it like this?

idk if u get what im saying

[lean otter]

Yea

What's a22 in your matrix

[sterile glacier]

the question or the answer?

[lean otter]

In the third matrix you've got

1 right?

[sterile glacier]

yeah

it's 1

[lean otter]

And in that a32 is?

[sterile glacier]

-3

[lean otter]

So what will be your new a32?

According to this

[sterile glacier]

6

HAHAHA

im stupid

damn i always do little mistakes

[lean otter]

Also you should just multiply 2nd row by 3 and add it to third

[sterile glacier]

ok thanks man i will redo the question

damn it sucks that i always make tiny little mistakes

really thankful

.solve

.close

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[sterile glacier]

.reopen

[safe radish]

✅

[sterile glacier]

anyone able to check my answer

[lean otter]

Check your y again

[safe radish]

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[mighty mango]

Whats the limit of (xsin2x+sin3x^2)/(tan^2(3x) + x^2)

[knotty pier]

Expansions?

[mighty mango]

What i did was divide the numerator and the denominator by x^2

[knotty pier]

Btw x tends to zero?

[mighty mango]

Yes

[knotty pier]

Ok then try expansions?

[mighty mango]

And i used the identity lim sinx\x=1 and lim tanx/x = 1 when x approaches zero

And i got 5/4

[knotty pier]

Yes

[mighty mango]

Im not asking “how to solve the question” i wanna know what i did wrong

[knotty pier]

Or just

Consider sinx is x and tanx is x

[lean otter]

You lost the active role?

[knotty pier]

U should be getting 1/2

[knotty pier]

Prolly

[mighty mango]

Yea and thats the answer

[knotty pier]

Send ur work

[mighty mango]

Ok but

[knotty pier]

Cuz dividing by x^2 is right approach

[mighty mango]

Ok

[knotty pier]

So send work and I'll check

[mighty mango]

Gimmie a sec to write everything in english

[knotty pier]

Yea ok

[mighty mango]

Now you gotta wait even longer for the pic to load ❤️

[knotty pier]

Np

Got plenty of time

[mighty mango]

Okay uhm

So sorry

For wasting your time

[knotty pier]

Nah fine even I want to waste time

[mighty mango]

Bc i realized i didnt square the tan

Its 5/10 indeed

[knotty pier]

Oh lol

Yea

[mighty mango]

Ok ty

.close

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[runic timber]

My question is that given the graph is equal to the derivative function of g(x), how would i estimate?

I know it should be roughly the x=4 and the area under the curve at and before the point

Would i estimate a left rieman sum?

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[opaque forge]

How do i find the roots of a 3th grade ecuation ?

[opaque forge]

Like this one

[obtuse plover]

grado -> “degree”

[opaque forge]

3th degree ecuation*

[safe radish]

@opaque forge Has your question been resolved?

[opaque forge]

Yeah

I'm trying toundersatnd what you just said

English isn't my first languege

So a and d

In this case

What i need to do whit them ?

1 and 2

And i get 2 ?

1 and 2

-1 , 1 ,-2 , 2

So I do f(1)

[safe radish]

@opaque forge Has your question been resolved?

[opaque forge]

And i use delta for X².....

So i do a×d

After find all the factors

And subsitute each factor in the exprasion to find a root

After i just ÷ the main f with the (X - n)

Right? @hasty mason

Thx

God bless you

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[candid ocean]

im a little confused

so the area of the flower garden is 42m^2 bc 7x6

[thin bridge]

yes

[candid ocean]

but what do i do now

not rlly

[thin bridge]

try drawing a diagram

[candid ocean]

ok but then what

i cant send the picture here

[thin bridge]

draw in paint

[candid ocean]

cant i do this without a diagram

[thin bridge]

you "can",
but having the diagram greatly helps you set up your equation

having a diagram is a good idea for the majority of geometry based problems

[candid ocean]

ok ty

.close

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[neon dawn]

how do u do this

[sonic bison]

How can I solve this integral?

[tepid walrus]

This channel is occupied please get another one :)

[sonic bison]

where

[tepid walrus]

Have you tried anything yet?

[neon dawn]

thanks

nope

[tepid walrus]

#❓how-to-get-help

[tepid walrus]

Well, you have y^2-x^3 and you know x=-1 and y=-2. What should you do with this information?

[neon dawn]

i keep doing it and get 3. the answer is 5 though. can you explain?

[tepid walrus]

It‘d be easier if I saw your work. That way I can see where you mess up

[neon dawn]

hold on

[neon dawn]

thats how i did it

[karmic hedge]

y^2**-**x^3

you can barely tell that the minus is bolded oops

[neon dawn]

ye aint that the same shit

4 - (-1)

ye i understand now my fault

.close

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[neon dawn]

can someone explain?

[split ether]

Do not confuse -x^2 with (-x)^2

[neon dawn]

why is it any different?

[split ether]

The first expression is the result of squaring x and changing its sign
The second expression is the result of changing the sign of x and squaring it

[neon dawn]

thanks. can you explain anoother question

[split ether]

Sure

[neon dawn]

answer is 10/3 but im not sure how

[split ether]

Have you tried anything so far?

[neon dawn]

yeah i can't figure it out

[split ether]

To isolate the x on the left hand side, it would be helpful to start by getting rid of that 1/3

Since it's being added, we can subtract -1/3 from both sides and get x/5 = 1 - 1/3

Can you simplify the right hand side?

[neon dawn]

Yes it is 2/3

Or 0.66

[split ether]

Right, so now x/5 = 2/3

[neon dawn]

Okay

[split ether]

Can you guess doing doing what could make the x/5 turn into x?

[neon dawn]

I'm not sure

[split ether]

The x is being divided by 5, what's the opposite of division?

[neon dawn]

multiply by 5

[split ether]

Try doing that to both sides of the equation x/5 = 2/3

[neon dawn]

x = 10/3

ahhhh

got it

thank you very much brother

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[wheat forum]

i try to subtract the area of the triangle from the area of the sector but it ends up negative

[marsh walrus]

whats the area of each?

[wheat forum]

i got 35.5 for triangle

and 17.5 for big sector

i mean small

[marsh walrus]

you probably want segment not sector?

sorry im mixed

[wheat forum]

ye

[marsh walrus]

,calc pi * 5^2 * 80 / 360

[flat frigate]

Result:

17.453292519943

[lean otter]

did u half the triangle

[marsh walrus]

yea this is what i wanted to check

[wheat forum]

no

[marsh walrus]

,w area of isosceles triangle

[flat frigate]

Results provided by WolframAlpha

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

Requested by janniku

[lean otter]

well u shouldnt have got above 25 to begin with i think

[marsh walrus]

im on mobile so i annoyingly cant draw

[wheat forum]

thats what i thought

[lean otter]

i got 12.31 for the triangle area, try working it out again

[dusky stag]

use 1/2 ab sin c for triangle

find sector area and subtr triangle area

[wheat forum]

what does sin c do

[lean otter]

u learnt trig yet?

[wheat forum]

i dont remeber

[lean otter]

sin/cos/tan ?

[wheat forum]

yeah i know that

[lean otter]

csc/sec/cot

[wheat forum]

yep

[dusky stag]

i got approximately 5.14320 as the answer

[lean otter]

sinh?

[wheat forum]

no

[lean otter]

good 😭

[dusky stag]

thats hyperbolic stop confusing it

[lean otter]

LMAO

[wheat forum]

i got it thx

.close

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[dusky stag]

anyway this is what i got

[safe radish]

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[arctic stone]

why is this neither odd or even

[safe radish]

Please don't occupy multiple help channels.

[arctic stone]

I don't understand the logic behind it

[gusty trench]

a function f(x) is odd if f(-x) = -f(x)

and a function f(x) is even if f(-x) = f(x)

notice that when you evaluate h(-x), you neither get -h(x) or h(x), so it can't be even or odd

[arctic stone]

okay I see thx neil

.close

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[red sonnet]

Hi, I just have a very simple question

I need to find all angles of each letter

Just wanna know if I solved this correctly

[heady lava]

a=46, correct?

[thin narwhal]

,rccw

damn wrong way

,rcw

[thin narwhal]

what are you finding

[heady lava]

and b=180-85-a, correct?

[red sonnet]

Sorry for messy writing

[heady lava]

and b+c=180, correct?

and b=d, correct?

[red sonnet]

I just need to find d, b, c and a

[red sonnet]

Yes

But not sure if a is right

[thin narwhal]

do you know vertical angle theorem

[red sonnet]

Yes

But I just was not sure

If it always applies

But now that you mention it I'm assuming it does

[thin narwhal]

it always applies i believe

[red sonnet]

So was the question I did correct?

[thin narwhal]

as long as theyre vertical angles

[red sonnet]

What do you mean?

[heady lava]

what do you mean?

[thin narwhal]

as long they are vertical angles VAT can be applied

[heady lava]

a equals to 46

[red sonnet]

A

Yeah I think everything should be correct

[heady lava]

ye if you know A, you can solve the triangle

thus you can solve the right thing

[red sonnet]

Yes ok

[thin narwhal]

b=48?

[red sonnet]

Yes

[thin narwhal]

r u sure

180 - 46 - 85

[heady lava]

i’m quite sure b equals to that

[thin narwhal]

yes thats the equation for b

r u sure b=48

[red sonnet]

Yes

As you said

[thin narwhal]

are you

100% certain

b = 48

[red sonnet]

Vertical angle

[thin narwhal]

im talking about angle b not angle a

[red sonnet]

Opposite from 46 is a

So a is 46

46 plus 85

Minus 180

[thin narwhal]

is

[red sonnet]

48

[thin narwhal]

no...

[red sonnet]

49!

[thin narwhal]

time to change your answers

[red sonnet]

Done

[thin narwhal]

since b is wrong, c and d is also wrong

[red sonnet]

Alright it's fixed

[thin narwhal]

Thats good

[red sonnet]

Yes

[safe radish]

@red sonnet Has your question been resolved?

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[void otter]

How do i solve for sp?

[torpid panther]

Hey there

[void otter]

?

[torpid panther]

Do you know how to start

[void otter]

nope

[torpid panther]

Ok wait

[void otter]

would it be (5x-4)/20=(3x+2)/15?

[safe radish]

@void otter Has your question been resolved?

[magic junco]

Good start, proceed.

[void otter]

would the answer just be 20/3

oh wait thats x nvm

ok thanks

.close

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[median vault]

can someone explain whats happening in this working out

im so confused

[west hedge]

P for probability?

[median vault]

this is the original question

[median vault]

huh

oh yeah

[dreamy elk]

$$P(A|B)=\frac{P(A\cap B)}{P(B)}$$

[lean otter]

do you know how to extend P(A / B) ?

[flat frigate]

Crystopher

[median vault]

ohh okay

whats the top part

[dreamy elk]

It is the probability of $A$ and $B$ happening at the same time.

[flat frigate]

Crystopher

[dreamy elk]

If $X>\frac{\pi}{4}$ ($A$) and $X<\frac{3\pi}{4}$ ($B$), then \ $P(A \cap B) = P(X>\frac{\pi}{4} \land X<\frac{3\pi}{4}) $ \
Which is the same as \
$P(\frac{\pi}{4}<X<\frac{3\pi}{4})$

[flat frigate]

Crystopher

[fickle pebble]

hi，guys I'm new to here.😁
I heard from other students, that there is lot of masters that will help with my question.can you guys help? thank you so much!!!😆

[safe radish]

@median vault Has your question been resolved?

[median vault]

alr thanks g

.close

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[rose knoll]

I need help with a 3D geometry question

[wheat cave]

how far have you gotten?

[rose knoll]

Nowhere, ig. I've been stuck on it for a while now

I've solved a similar problem where the plane is rotated by 90 degrees, it was easier to deal with

So far what I understand is

The plane is rotated along the x-y plane by angle alpha

Need a point and direction

But idk how😭

[rough storm]

a plane is described by its normal vector and a point on that plane

@rose knoll

[rose knoll]

Yup

[rough storm]

what point is always on this plane :3

no matter the l or m values

and is also located in the z=0 plane

[rose knoll]

All the points in the line of intersection of original plane and z=0 (?)

[rough storm]

mhm

but whats like, the most easiest to think about

[rose knoll]

(l, m, 0)
? I'm not sure

[rough storm]

well its not easy to picture rotating a vector at (l,m,0) is it

and l,m,0 might not even be a point on this plane, that would require l^2+m^2=0

[rose knoll]

Aha

[rough storm]

well, it wouldnt

[rose knoll]

It wouldn't be

[rough storm]

what point WOULD be on this plane (don't think too hard about it) whats like, the easiest x and y value that you could plug in and the equation would hold

[rose knoll]

The plane after rotation right?

This is what I'm picturing

I can't tell😭

[rough storm]

yes

what line must this plane cross through

it crosses through two lines

well, it 'contains' two lines

[rose knoll]

The line of intersection, and the normal line of original plane

Wait

No

[rough storm]

two lines lie completely within this plane

one of them is the line of intersection

[rose knoll]

It wouldn't have the normal in it

Projection of the normal on this plane?

[rough storm]

you're thinking about it too hard. think about how the z-axis, x-axis, and y-axis relate to this plane

note that z is a 'free variable' too, i.e. it can be anything and the equation will still hold

[rose knoll]

Oh right... It could be anything

I'm more confused... I think I'm picturing something wrong

Like is the origin involved

😭

[rough storm]

INDEED it is

you can solve for the line you're rotating about too

[rose knoll]

So it's like this, extended about the z axis

[rough storm]

so since we're rotating about the line of intersection, which contains the origin, we're safe to include the origin as the point for our new plane

and then simply rotate the normal vector by alpha to obtain our new normal vector for our new plane

[rose knoll]

Aha!

[rose knoll]

How'd we find that normal?

I need to review vector rotation

Thank you so far, I learned something

[rough storm]

a vector is defined by $N \cdot P=a(x-x_0)+b(y-y_0)+c(z-z_0)=0$ where $N=<a,b,c>$ and $P=<x-x_{0},y-y_{0},z-z_0>$

[flat frigate]

🫎 Moosey 🫎

[rough storm]

N is normal vector

P is a general vector pointing out from a point in the plane

https://tutorial.math.lamar.edu/classes/calciii/eqnsofplanes.aspx

[rose knoll]

I'm struggling w the normal vector

What is it after rotation?

I looked it up
It seems it has something to do with matrix rotation?

[rose knoll]

Got this

[rough storm]

are you familar with spherical coordinates?

[rose knoll]

I recently learned about them

[rough storm]

you can scale this normal vector to be a unit vector and treat it as a point on a unit sphere

[rose knoll]

WOAH

[rough storm]

it doesn't have to be on unit sphere

it might be easier if its not

[rough storm]

this is where i imagine the sqrt(l^2+m^2) comes from

as this is the normals magnitude currently

[rose knoll]

Can this not be done by basic trig? Since z is not involved

I tried something but it got really nasty

[rough storm]

you could do it with some trig yes, if you're clever with how you place circles

[rose knoll]

I'm gonna try again

[safe radish]

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[candid ibex]

angles

[candid ibex]

perpendicular bisectors

[torpid panther]

?

What about them

[candid ibex]

how do they wrk?

i forgot💀

[torpid panther]

Which

Angles?

Or perpendicular

[candid ibex]

basic ideas in geometry

perpendicular

[torpid panther]

Do you have an example of a question

Will be easier to explain

[candid ibex]

hold up

[spice wing]

he is asking how to draw a perpendicular bisector maybe

And angle bisector maybe

[torpid panther]

Perpendicular bisector

Must have an angle of 90 degree

And must intersect at midpoint

[candid ibex]

ok

bro

can you explain these 2 topics

[torpid panther]

[candid ibex]

i missed todays day of school

[torpid panther]

Umm

[candid ibex]

[torpid panther]

Do you know what quadrilaterals are

[candid ibex]

i have past knowlege

[torpid panther]

Sorry I’m not rly here to teach whole topics 😓

Unless u have specific questions

[candid ibex]

ok jsut say what is "directions turns"

[safe radish]

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[timber turret]

can some one draw it and explain it

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[timber turret]

.reopen

[safe radish]

✅

[timber turret]

<@&286206848099549185>

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[candid verge]

I wanted to see if there was a way to find the exact value of alpha (I know you could just use newton rhapson etc)

[candid verge]

but whilst usually you would use l'hopital's since ln0 and 1/0 are undefined when ur evaluating 0^0

u cannot use it in this case (and if you try you get the wrong answer)

so is there any way to do this or is it not possible

[safe radish]

@candid verge Has your question been resolved?

[candid verge]

<@&286206848099549185>

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[fathom jewel]

,, \sin X \leq y \stackrel{?}{\implies} X \in [-\pi, -\arcsin y] \cup [\arcsin y, \pi ]

[flat frigate]

add on is unacceptable

[toxic stratus]

i'd say no

you are missing assumptions on y and X

[fathom jewel]

YAH

MY PROFESSOE IS WRONG

Ahahah

[toxic stratus]

[fathom jewel]

I HATE HIM

[toxic stratus]

[fathom jewel]

I AM COOKED FOR PROBABILOTY THEORY

[toxic stratus]

this isn't even probability theory

[desert pasture]

plot twist, snow is your professor

[fathom jewel]

I WILL SHOW CONTEXT

[toxic stratus]

no but this specific thing

is just like

idk

highschool trig

[hard crest]

eg X can be 5 billion and change

[fathom jewel]

X ~ U(-π,π)

[hard crest]

oh

[toxic stratus]

okay that's kinda important to know

[hard crest]

garbage in garbage out

[toxic stratus]

and also what y is

[hard crest]

u are a helpful you should know context is important

[toxic stratus]

like if you add the assumptions that -pi < X < pi, and that -1 < y < 1, then you can say a lot more

[hard crest]

anyway yes this is high school trig

[fathom jewel]

I am too dumb

[hard crest]

draw a sketch of sin x and of y = Y

[toxic stratus]

if you don't draw a sketch you're kinda going in blind

[fathom jewel]

I am trying so hard for 2 hours ;-;

[fathom jewel]

forgive me 😭

[toxic stratus]

it feels like you're trying to bash this with pure symbol pushing

you need to just approach it geometrically

[fathom jewel]

ok 🫡

[toxic stratus]

how do i share a desmos graph on phone

[hard crest]

screenshot

[toxic stratus]

do this

make it yourself on desmos

[fathom jewel]

yes

high school trig but my professor failed twice ultra ☠️

[fathom jewel]

I am curious @dull sequoia

[toxic stratus]

it's pretty wrong

[fathom jewel]

I am mid class currently I will try later

Thanks my pookies

.solved

[safe radish]

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[toxic stratus]

👍

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[jovial niche]

This is my question in the start of an ODE course (ordinary differential equations) dy/dx = (xy +y)/(x+xy), I am asked to find the funciton y(x). Here is what I tried, the upper part got me a wrong answer (now that I think about it, it's silly of me to do an integral according to dy since it is a funciton of x so the rules are different than when integrating using a regular variable). The lower part feels like it's going nowhere.

[jovial niche]

sec I will post the picture

[lean otter]

dy/dx = (xy+y)/(x+xy)

thats ur question?

[jovial niche]

yes

[lean otter]

first id recommend separating y and x from eachother

we can just factorise for that

y(x+1)/x(y+1)

then from there we can separate into 2 fractions

and solve the DE

well find a general solution

[jovial niche]

oh so (y/(y+1))*((x+1)/x)?

[lean otter]

yes

and from there do u have a better idea on how to solve it?

[jovial niche]

then I divide by (y/(y+1)) and mulitply by dx I think would be the next step

and integrate from there

alright thanks

.close

[safe radish]

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[lean otter]

👍

[jovial niche]

I closed it, yet for some reason it still in occupied?

.close

.close

[lean otter]

takes time dw

[safe radish]

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[rare canyon]

how do i do test convergence for the integral sqrt(x^3+1)-sqrt(x^3) from 1 to infinity?

[safe radish]

@rare canyon Has your question been resolved?

[safe radish]

@rare canyon Has your question been resolved?

[safe radish]

@rare canyon Has your question been resolved?

[rare canyon]

Yes

[safe radish]

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[fathom jewel]

.

lol

[fathom jewel]

How do I do this

uhm

[cerulean crane]

Just post the math question you have and what you've tried

[fathom jewel]

i know

[fathom jewel]

so i was stuck here

and found out this

So basically

let me rewrite the context

$X \sim U(-\pi, \pi), \quad Y:=\sin X\$
What is the cdf $F_Y(y) = : ? :\text{ where } y \in \mathbb{R}\$

So basically

\begin{align*}
F_Y(y)
&= P(Y \leq y) \
&= P(\sin X \leq y) \
&= P \big ( { \frac{\pi}{2} \leq X \leq \pi : : : \pi - \arcsin y \leq X \leq \pi } \
&\cup { -\pi \leq X < \frac{\pi}{2} : : : (\pi \leq X \leq \arcsin y) \land (\sin X \leq y)} \big )
\end{align*}

[flat frigate]

add on is unacceptable

[fathom jewel]

actually I tried to do 3 regions

but I keep messing up

I kinda have to subtract the yellow piece and that is my issue

Ok the yellow piece is $-\pi \leq X \leq -\arcsin y -\pi$

[flat frigate]

add on is unacceptable

[fathom jewel]

So

\begin{align*}
F_Y(y)
&= P(Y \leq y) \
&= P(\sin X \leq y) \
&= P \bigg ( \bigg { \abs{x} < \frac{\pi}{2} : : : X \leq \arcsin y \bigg } \
&\cup \bigg { \frac{\pi}{2} \leq X < \pi : : : \pi - \arcsin y \leq X \leq \pi \bigg } \
&\cup \bigg { -\pi \leq X < -\frac{\pi}{2} : : : -\pi \leq X \leq -\arcsin y \bigg } \
&\setminus \bigg { -\pi \leq X < -\frac{\pi}{2} : : : -\pi \leq X \leq -\arcsin y -\pi \bigg } \bigg ) \
&= \int_{-\frac{\pi}{2}}^{\arcsin y} \frac{1}{2\pi} \dd u + \int_{\pi - \arcsin y}^{\pi} \frac{1}{2\pi} \dd u + \int_{-\pi}^{-\arcsin y} \frac{1}{2\pi} \dd u -\int_{-\pi}^{-\arcsin y -\pi} \frac{1}{2\pi} \dd u \
&= \frac{1}{2\pi} \left ( \int_{-\frac{\pi}{2}}^{\arcsin y} :\dd u + \int_{\pi - \arcsin y}^{\pi} : \dd u + \int_{-\pi}^{-\arcsin y} : \dd u -\int_{-\pi}^{-\arcsin y -\pi} : \dd u \right ) \
&= \frac{1}{2\pi} \left [ \left ( \arcsin y - \frac{-\pi}{2} \right ) + \left ( \pi - (\pi - \arcsin y) \right ) + \left ( -\arcsin y - (-\pi) \right ) - \left ( -\arcsin y -\pi - (-\pi) \right ) \right ] \
&= \frac{\frac{3\pi}{2} + 2\arcsin y}{2\pi} \
&= \frac{3}{4} + \frac{\arcsin y}{\pi}
\end{align*}

,w 1/(2pi) * ((arcsin(y) - (-pi/2)) + (pi - (pi-arcsin(y))) + (-arcsin(y) - (-pi)) - (-arcsin(y)- pi -(-pi)))

[flat frigate]

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Requested by adoniszgks

add on is unacceptable

[fathom jewel]

damn I am off

instead of 3/4 it should been 1/2

ARERHH

[novel stag]

have you learned double integrals?

[fathom jewel]

ye

[novel stag]

shouldnt be too hard to set up for the yellow part

[fathom jewel]

I just did the yellow part

[novel stag]

wait

what area do you want to find

[fathom jewel]

sinX <= y I need to transform this in terms of X which results to cases

$X \sim U(-\pi, \pi), \quad Y:=\sin X\$
What is the cdf $F_Y(y) = : ? :\text{ where } y \in \mathbb{R}\$

[flat frigate]

add on is unacceptable

[fathom jewel]

this is what I am figuring out

[fathom jewel]

and that's my progress

because my professor did it twice wrong so i need to kinda get on my own

[agile rapids]

Hello is anyone around to help a brother in need

[fathom jewel]

!occupied

[safe radish]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[fathom jewel]

np

[fathom jewel]

i think I did something wrong and found it

damn why

\begin{align*}
F_Y(y)
&= P(Y \leq y) \
&= P(\sin X \leq y) \
&= P \bigg ( \bigg { \frac{\pi}{2} \leq X < \pi : : : \pi - \arcsin y \leq X \leq \pi \bigg } \
&\cup \bigg { -\pi \leq X < -\frac{\pi}{2} : : : -\pi \leq X \leq \arcsin y \bigg } \
&\setminus \bigg { -\pi \leq X < -\frac{\pi}{2} : : : -\pi \leq X \leq -\arcsin y -\pi \bigg } \bigg ) \
&= \int_{\pi - \arcsin y}^{\pi} \frac{1}{2\pi} \dd u + \int_{-\pi}^{\arcsin y} \frac{1}{2\pi} \dd u -\int_{-\pi}^{-\arcsin y -\pi} \frac{1}{2\pi} \dd u \
&= \frac{1}{2\pi} \left ( \int_{\pi - \arcsin y}^{\pi} : \dd u + \int_{-\pi}^{\arcsin y} : \dd u -\int_{-\pi}^{-\arcsin y -\pi} : \dd u \right ) \
&= \frac{1}{2\pi} \left [ \left ( \pi - (\pi - \arcsin y) \right ) + \left ( \arcsin y - (-\pi) \right ) - \left ( -\arcsin y -\pi - (-\pi) \right ) \right ] \
\end{align*}

found

[flat frigate]

add on is unacceptable

[fathom jewel]

,w 1/(2pi) * ( (pi - (pi-arcsin(y))) + (arcsin(y) - (-pi)) - (-arcsin(y)- pi -(-pi)))

[flat frigate]

Results provided by WolframAlpha

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Requested by adoniszgks

[fathom jewel]

no I cant find the mistake

[toxic stratus]

oh my god

what is all this

[fathom jewel]

hi

[toxic stratus]

there are two cases you need to consider

[fathom jewel]

hayley said it's high school but i am making a fuss probably

[toxic stratus]

y >= 0 and y < 0

[fathom jewel]

i thought i do the cases in terms of x

[toxic stratus]

Assume $y \ge 0$. Then for $X \in (-\pi, \pi)$,
[ \sin(X) \le y \Iff X \in (-\pi, \pi) \setminus (\arcsin(y), \pi - \arcsin(y)) ]

[toxic stratus]

its this picture

[fathom jewel]

If y >= 0 then sin(X) <= y is true for X in [0,pi]

[toxic stratus]

no

[fathom jewel]

oh no -pi to 0

[toxic stratus]

if you take y = 1/2, sin(pi/2) = 1

you just solve sin(X) = y

X = arcsin(y) or pi - arcsin(y)

and outside of the interval (arcsin(y), pi - arcsin(y)), sin(X) will be below y

the red curve falls below the blue horizontal line in the blue highlighted region

[fathom jewel]

gimme some i am slow

,, \sin X = y \iff \left ( X = \arcsin y, : \abs{X} < \frac{\pi}{2} \right ) \text{ or } \left ( X = \pi - \arcsin y, :\frac{\pi}{2} < X < \pi \right )

[flat frigate]

add on is unacceptable

[toxic stratus]

i mean sure you can say that

[fathom jewel]

i just need reassuranc

but what about

-pi to -pi/2

[toxic stratus]

assume 0 <= y <= 1 for now

[fathom jewel]

ok

[toxic stratus]

the equation sin(X) = y has two solutions

X = arcsin(y) and X = pi - arcsin(y)

[fathom jewel]

ok i accept

[toxic stratus]

to find where sin(X) <= y, you just look on either side of those solutions

the point is that you want to see when the red curve falls below the blue line

and by solving sin(X) = y, you find those two intersection points that i've marked

at this point, you can just inspect graphically or something to see that to the left of arcsin(y), sin(X) < y

and to the right of pi - arcsin(y), sin(X) < y

[fathom jewel]

0 <= y <= 1

for 0 <= y <= pi/2 we get the first solution arcsin(y) and the second is is between pi/2 <= y <= 1 so we basically

arcsin(y) -> -arcsin(y) -> -arcsin(y) + pi

[toxic stratus]

hold on

[fathom jewel]

that's how i kinda get it

[toxic stratus]

no thats really sus

0 <= y <= pi/2

[fathom jewel]

yea

mb

i actually meant x

[toxic stratus]

okay

[fathom jewel]

damn

[toxic stratus]

sure

[fathom jewel]

yea it makes sense

[toxic stratus]

if you want to solve your trig equations that way then fine

so the conclusion is

for $0 \le y \le 1$ and $X \in (-\pi, \pi)$,
[ \sin(X) \le y \textqq{is equivalent to} X \in (-\pi, \pi) \setminus (\arcsin(y), \pi - \arcsin(y)) ]

[fathom jewel]

i am trying to get it graphically

[toxic stratus]

other way round

i kinda butchered that drawing but it curves that way

[fathom jewel]

isee

[toxic stratus]

although the graph of arcsin(y) isnt really relevant

sin isnt injective so its not so helpful to be staring at that graph

[toxic stratus]

this is the one you want to be thinking about

[fathom jewel]

So to summarize the first approach.

We kinda just assumed 0 <= y <= 1.

Generally there are two solutions within in that, and the way to get them is like always doing that trig stuff with arcsin(y) and subtracting arcsin(y) from pi.

And now we want what

[fathom jewel]

how does the blue area happen

[toxic stratus]

because of this

to the left of arcsin(y), sin(X) < y
to the right of pi - arcsin(y), sin(X) < y

[fathom jewel]

Oh

Oh

it's below the blue line

[toxic stratus]

so the solution to sin(X) <= y is -pi < X < arcsin(y) or pi - arcsin(y) < X < pi

[fathom jewel]

the blue line represents 0 <= y <= 1 right?

[toxic stratus]

yes

[fathom jewel]

damn i am so lost ok

good

idk why but this was actually trivial i guess

or not

[toxic stratus]

this is really solving trig inequalities

has nothing to do with integration or probability or whatever

[fathom jewel]

this makes now totally sense

hmm yeah

[toxic stratus]

the story for -1 <= y < 0 is pretty much the same

[fathom jewel]

now that i think of it i think i actually never done trig inequalties

[fathom jewel]

i got a feeling what it is

[fathom jewel]

ye

hold on

arcsin(y) and -pi -arcsin(y) would be the two intersections

and now it's $X \in [-\pi -\arcsin y, \arcsin y]$

[toxic stratus]

[]

[flat frigate]

add on is unacceptable

[toxic stratus]

so now its pretty simple to calculate the probabilities

[fathom jewel]

this is one integral

[toxic stratus]

X is uniform, so don't do any integrals

all you have to do is look at the lengths of the intervals

[fathom jewel]

what how

this is always what we done in the classes

[toxic stratus]

for $0 \le y \le 1$,
[ P(\sin(X) \le y) = P(X \in (-\pi, \pi) \setminus (\arcsin(y), \pi - \arcsin(y))) ]

[toxic stratus]

but because X is a uniform random variable

you can calculate its probabilities directly by looking at the lengths of the intervals

[fathom jewel]

isn't it just a short cut basically?

because

[toxic stratus]

[2\textwidth ][ P(X \in (-\pi, \pi) \setminus (\arcsin(y), \pi - \arcsin(y))) = \f {\text{length of } (-\pi, \pi) \setminus (\arcsin(y), \pi - \arcsin(y))} {\text{length of } (-\pi, \pi)} ]

[fathom jewel]

1/(b-a) is a constant that we would pull out

if we integrated

[fathom jewel]

and then it's only the bounds

[toxic stratus]

theres not really much point doing the integral here

you can just stare at the interval

and know the answer

[fathom jewel]

ok ok i am just trying to understand

what is length of

[toxic stratus]

intuitively the length of the interval (a, b) is b - a

[fathom jewel]

I SEE

so it is a short cut

[toxic stratus]

this is how the uniform distribution is defined in the first place

you should be able to see this without doing integration

[fathom jewel]

ok (dont blame if my professor taught us by doing integration)

[toxic stratus]

the whole point of the uniform distribution is that its just looking at the length/area/volume present

[fathom jewel]

So it's basically

1/2 + arcsin(y)/pi

[toxic stratus]

probably

(2pi - (pi - 2arcsin(y))) / 2pi

[fathom jewel]

[pi + 2arcsin(y)] / 2pi

[toxic stratus]

whatever that is

yeah

[fathom jewel]

YOU ARE A KING/QUEEN

[toxic stratus]

then for -1 <= y <= 0, its the same agian

[fathom jewel]

this makes sosense

[toxic stratus]

for $-1 \le y \le 0$,
[ P(\sin(X) \le y) = P(X \in [-\pi - \arcsin(y), \arcsin(y)]) ]

[fathom jewel]

hayley was so right it was high school

I think what we did was we generally defined the cdf as $$F_X(x) = P(X \leq x) = \int_{-\infty}^x f_X(x) : \dd x$$
I know bad notation sigh.

[fathom jewel]

but i see this works faster

idk if that makes sense to you

oh no mb

[toxic stratus]

while thats true its not always the quickest way to do things

[flat frigate]

add on is unacceptable

[fathom jewel]

oh it was right?

[toxic stratus]

f_X(t) dt preferably

but i mean thats a true statement

[fathom jewel]

YEESSSS

SO COOL

i still cant tell how my professor screwed this up so bad

but thank youu

.solved

[safe radish]

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[safe radish]

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• Show your work, and if possible, explain where you are stuck.
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[cerulean crane]

!status

[safe radish]

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

[hushed flume]

1

[cerulean crane]

Well, do you know how $a^{-b}$ is defined

[flat frigate]

خرشوف

[hushed flume]

1/a^b

[cerulean crane]

Right, so how can we express $\frac{x^ay^{-b}}{x^{-a}y^b}$ using this

[flat frigate]

خرشوف

[sweet dove]

can't we write this as two fractions, then use power rule ?

try using the fact that (x^n)/(x^p) = x^(n-p)

[hushed flume]

Uhh

[sweet dove]

btw can u show the forms they give u ?

[sweet dove]

okay

did u try to use the power rules ?

its the only thing u need to do

[sweet dove]

and btw:

[cerulean crane]

This is right, yeah

Now how can we simplify it

[cerulean crane]

Oh you got it, cool

[safe radish]

@hushed flume Has your question been resolved?

[hushed flume]

Idk how

[hushed flume]

No I used the answer key

[cerulean crane]

Well the numerator and denominator are of the form $\left(\frac{1}{a}\right)b$

[flat frigate]

خرشوف

[cerulean crane]

How do we simplify something like that

[hushed flume]

Hmmmm

Multiply

[safe radish]

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[main cipher]

What are these kinds of problems called?

[main cipher]

I want to be able to look them up for practice

[faint seal]

,rccw

[faint seal]

uhh, looks like just simple algebraic simplification problems to me

you won't "find" these on the internet

it's like asking for problems like 10/20 or 3+4/7

it's just the extra "i" added, it's still just basic arithmetic

[main cipher]

I'm pretty sure there was a name for this skill we were practicing

[ruby ether]

complex number problems

[main cipher]

bingo! thank you!

.close

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[dusk rock]

.

Hi

[magic junco]

Any questions?

[dusk rock]

Find the points such that y=2x is the tangent line to the function ⬆️

Am i doing this right?

This means that all points x in the function will make y=2x?

[thin bridge]

no

[dusk rock]

I have no idea how to proceed to try finding the values

[thin bridge]

x isn't the same as x_0
the simpler approach would be to set derivative= slope of tangent line

and for extra rigor confirm the point of the curve and tangent line is the same at that location

[dusk rock]

So I try equalizing 1/x +1 (the slope of tangent line) = 2x

I found it thankss

[thin bridge]

no

derivative is 1/x + 1
slope if tangent line is just 2

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[mental brook]

Let $\alpha \in \mathbb{R}^*,, p:\mathbb{H} \to \mathbb{D} \setminus {0}, , p(z) = e^\frac{2 \pi i z}{|a|}$. I want to show that $p$ is a covering map but I dont't know how to make this. First, clear $p$ is continuous and surjective. I think I need to start with an $y \in \mathbb{D} \setminus {0}$ and take an open disk $D \subset \mathbb{D} \setminus {0}$ to be a neighbourhood of $y$. Now, because disk is open and connected, there exists a holomorphic branch of logarithm in $D$. It's ok this start? How I can continue? Thanks!

[flat frigate]

Tibi

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@mental brook Has your question been resolved?

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@mental brook Has your question been resolved?

[crisp minnow]

Probably better luck in #real-complex-analysis

[mighty mango]

@mental brook

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[dry raptor]

I’m trying to create a function that’s basically to exceed Infinity (1.79e308 / 2^1024), I supposed it could be a simple two variable function where x is the growing function influenced by y (y is a fixed variable)

Suppose in a non-scientific notation, x is 100, the scientific notation makes x equal to 1.00e2,, or in a logarithmic scale, just e2—incrementing x by y (equal to 30), this would make x approximate e2.724...

I’ve tried f(x + 1 * [log10 of {x + 1} * y] ) which seemed to follow a natural logarithmic growth before going exponential. I’ve also tried making the base of log exponential which basically halted the function in a logarithmic scale. (I noted these by using a reference variable which made another seperate function f(log10 of [z + y] ) where z simply is a linear growth with y’s influence)

[dry raptor]

||im not sure if this server is the correct one to ask since this is more of a coding question||

[crystal palm]

What should the function do? "Exceed infinity" sounds like you want it to approach infinity

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[lean otter]

what is the amplitude and the period?

[gusty trench]

period is the time it takes for the wave to repeat itself

so a good strategy is to figure out what time a peak or valley occurs and look for the next one

[lean otter]

i know that, but its not an integer number

[gusty trench]

the period here looks like about 5?

it's hard to tell exactly since you only have gridlines at each integer

but there's a valley that occurs a bit before 2, maybe around 1.7 or 1.8, and the next one happens around 6.7 or 6.8

[lean otter]

ok, i I understood.
and how i get the amplitude?

[gusty trench]

amplitude is the vertical distance from the top/bottom of the wave to the central "equilibrium"

[buoyant shadow]

it's integer

[gusty trench]

it can also be found by taking half of the distance between a peak and valley

[buoyant shadow]

look at 3 and 8

[lean otter]

so, is 6 or 3?

[gusty trench]

3

[lean otter]

why

if it goes to -3 to 3

[gusty trench]

0 is like the "equilibrium value"

and it goes 3 above and 3 below

that's why you either look at the displacement from the equilibrium, or half the range

and if you think about how this comes from an equation

if this curve was something like 3cos(x) + ...

or 3sin(x) + ...

the cosine and sine terms are always between -1 and 1

so to get oscillation that goes either 3 above or 3 below, you multiply them by 3

that's why the amplitude is half the distance from the top and bottom

[lean otter]

ohh, i understood.
thank you so much

.close

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[thin bridge]

wdym by "remember"

[buoyant shadow]

you don't need to

if a question asks you to recreate it, it will give you enough time to calculate

[thin bridge]

the sides are all 1s
each number is the sum of the two above it

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[proper vortex]

Can someone explain this

[ripe cove]

it comes out to be 1/sec^2 + 1/cosec^2

you cannot put values where sec and cosec are zero

[ripe cove]

by dividing both sides by sec^2 multiplied by cosec ^ 2

I hope that is a simple and good enough explanation :).

so the domain is all values except those where they are zero

since 1/0 isnt defined

[proper vortex]

You mean 1/sin^2 +1/cos^2

[ripe cove]

for finding domain just look for stuff that causes problems such a dividing by zero, negative values in logarithm or in root

[ripe cove]

wait i didnt think of it tht way

it's written as 1 = 1

1/cosec^2 + 1/sec^2 = sin^2 + cos^2 = 1

[proper vortex]

With b and c both sides are equal

[ripe cove]

but it is asking for domain not values of phi

so i still think my solution is valid

[proper vortex]

Are you solving the lhs separately

[ripe cove]

cuz think of it this way, when we divide both sides by cosec^2 multiplied by sec^2

[proper vortex]

Yessir
And jn options it says ≠

[ripe cove]

we cant divide by zero

so we need to remove those values

only then 1 = 1

so we need to remove the values where either cosec or sec phi is zero

it is also another nice way of looking at things :)

[proper vortex]

Aight lemme

I'm lost

[ripe cove]

aw

where was the last place where things made sense to you

[proper vortex]

Hmm well

[proper vortex]

Sorry this makes no sense

[ripe cove]

ok so what is did

os that the sec^2 cosec^2 on the right side

i took it from right side

and put it in left side

[proper vortex]

So you mean this right

[ripe cove]

does tht make sense?

[proper vortex]

OH WAIT ONG

[ripe cove]

lol u placed the sec^2 into numerator

[proper vortex]

wait wait ok j see

[ripe cove]

it is also in denominator

[proper vortex]

Yeah I thought you said multiply

[ripe cove]

sorry :)

i meant divide both sides by sec^2 * cosec^2

[proper vortex]

1/cosec^2 + 1/sec^2 =1

[ripe cove]

yeah

[proper vortex]

Totally my error itsok

[ripe cove]

but we need to remove the values where either cosec or sec comes out to be zero

because then it will become not defined

[proper vortex]

Yeah

[ripe cove]

as we can't rly divide by zero

so was my service satisfactory :)

[proper vortex]

I get that vut😭

[ripe cove]

?

[proper vortex]

SECANT IS ZERO AT 90

[ripe cove]

it is also zero at 270

and so on

[proper vortex]

And cosec is zero at 180

[ripe cove]

cosec is also zero at 0

and 360 and so on

[proper vortex]

How is it a

[ripe cove]

secant is zero at all the values of n pi / 2 where n is an integer

[proper vortex]

Yep

[ripe cove]

and cosec is zero at all the values of n pi

[proper vortex]

Mhm

[ripe cove]

so remove those values

and you will get the required domain