how is my guy getting 200
#help-17
silver tusk
silver tusk
yed
silver tusk
yes
vast shale
and whats tan(180-115) ?
vast shale
no
silver tusk
Isn'titinquadrant
vast shale
try to get tan 200 from tan 20
vast shale
oh :p
vast shale
yeah
silver tusk
now what
silver tusk
yes
silver tusk
try
silver tusk
tan 180+20
vast shale
yep
vast shale
what formula
silver tusk
tan(a+b)
silver tusk
?
vast shale
why do you want to expand ?
vast shale
why can't you ?
vast shale
wdym "try" ?
vast shale
its not equal to tan (90)
it is
silver tusk
1+1
vast shale
tan(45+45)=tan90
nova iris
tan ( 45+45)
2/1-1 = undefined
vast shale
I have to leave,could you take it from here @nova iris?
nova iris
Ya sure
vast shale
thank you
nova iris
Can you repost the question pls
nova iris
Tan (a-b) = tan a - tan b / 1 + tana tanb
silver tusk
nvm
silver tusk
Tan (a-b) = tan a - tan b / 1 + tana tanb
ye
nova iris
It shoudve been that
nova iris
He did a mistake
silver tusk
how
nova iris
That's the literal formula
nova iris
Convert into sine cos
nova iris
Yes and the denominatior
silver tusk
sin65cos20 - sin20cos65/ cos65cos20
silver tusk
hi @nova iris
silver tusk
sinAcosB - sinBcosA
nova iris
It's +
silver tusk
ye
nova iris
See the numerator is now form sin(a-b)
silver tusk
yee
silver tusk
yes
nova iris
So what is the number sin45/cos45
silver tusk
tan45
nova iris
Which is tan 45
silver tusk
1
nova iris
And that's equal to 1
silver tusk
correct
silver tusk
Therss your answer
the question isn't done completely yet
nova iris
What's left to do?
silver tusk
tan 200 tan 115 = 1+ tan 200 + tan 115
nova iris
What is tan 200?
silver tusk
180+20
silver tusk
90+25
silver tusk
yes
nova iris
If it's tan (90+x) = -cot x
silver tusk
ye
nova iris
This rearranging is clear?
This rearranging is clear?
silver tusk
This rearranging is clear?
wait
nova iris
Here's the thing if it's sin(n*pi/2 +x) and n is odd it converts
So sjn to cos tan to cot etc
Here n was 1 ie odd
So it becomes cos
And tan cot
silver tusk
in first and third it stays same?
no I mean opposite
Iamconfused
which quadrants does sin become cos
nova iris
But it really depends on how you do it
You can also change it there
By doing 3pi/2 - x in 3rd quad
To convert
silver tusk
This rearranging is clear?
ye clear
nova iris
Then Now divide rhs on both sides you get your answer
Put it in tan(a-b) you proved it
Write presuming this equality holds this must be true
And we know tan45 = 1 so that equation is correct
nova iris
Anytime dude
silver tusk
.close
vocal sleetBOT
vocal sleetBOT
Available help channel!
dry bone
Can anybody help me with this?
When the area limited by the curve y=x(A-x) and the x-axis is S and the area limited by the curve y=x²(A-x) and the x-axis is S1, find the value of A so that S=2S1. 0<A≤1
desert hornet
Drawing a rough figure first might be helpful
dry bone
The answer is A=1 but I don't understand how it got to that result
desert hornet
What region do each of the areas represent?
dry bone
true grail
ok i made a graph
true grail
right so what u wanna do is form the area under each curve in terms of A seperately
desert hornet
Solve the integrals and set S=2S1 and solve for A
true grail
and then combine those expressions with $S = 2 S_{1}$ and use that to find A
dry bone
I think I could use
y-f(a)=f'(a)(x-a)
but I dont know how
twin meteorBOT
Out Of Nosh
true grail
first step is to find where the graph crosses the x-axis on the right in terms of A for each curve
so that u have the non-0 limit on your integration
desert hornet
I think I could use
y-f(a)=f'(a)(x-a)
but I dont know how
Why would you need this?
This is the equation of a tangent
dry bone
Couldn't I use it to find the slope of any of the functions at the point (A, 0)?
desert hornet
Why would you want the slope?
dry bone
couldnt be useful in something?
true grail
not really, no
dry bone
Okay, but we have the areas and with them how can we find the value of A?
I think I managed to solve it
I equated the functions to find the intersections, so it resulted in x=0, A, 1
Can I simply say that A=1 since x=A and A>0?
true grail
Can I simply say that A=1 since x=A and A>0?
yes
dry bone
okay, thank you so much
vocal sleetBOT
@dry bone Has your question been resolved?
hollow maple
what is the statement reason for these two problems?
vocal sleetBOT
what is the statement reason for these two problems?
hollow maple
shadow raft
!status
vocal sleetBOT
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
7
sharp comet
what does the given say?
vocal sleetBOT
@hollow maple Has your question been resolved?
outer warren
what's your status that isn't described in options 1 - 6
chilly locust
$\lim_{x \to + \infty } f(x) = a ; ; ; and ; ; ; \lim_{x \to - \infty } f(x) = a \iff \lim_{x \to \infty } f(x) = a$
vocal sleetBOT
$\lim_{x \to + \infty } f(x) = a \; \; \; and \; \; \; \lim_{x \to - \infty } f(...
twin meteorBOT
ǝʞnsᴉɐꓷ
chilly locust
Is it accurate if these limits exists?
outer warren
no
arrow doesn't work both ways
limit to inf by itself tells you nothing about limit to -inf
viral plover
vocal sleetBOT
vocal sleetBOT
Available help channel!
tawny knot
how can i find the inverse laplace transform of 1/(s+2)^2 using the bromwich integral?
vocal sleetBOT
how can i find the inverse laplace transform of 1/(s+2)^2 using the bromwich int...
tawny knot
apparently i have to do a substitution here for cauchys formula to work but i dont understand it exactly
oh nvm i found it
.close
vocal sleetBOT
vocal sleetBOT
Available help channel!
waxen hawk
Ahhh
mellow oyster
... do you have a question
waxen hawk
Yes
vocal sleetBOT
This channel is only for on-topic discussion. Please take casual conversation to #discussion or #chill.
urban edge
which euler number
waxen hawk
I hate Euler number
urban edge
there are like ten million euler numbers
waxen hawk
I’m so mad
urban edge
no
mellow oyster
what do complex roots of unity have to do with euler's number?
urban edge
oh right
waxen hawk
My heart is kidnapped by his beauty
urban edge
you should learn about mod-arg form and de moivres theorem before you can fully understand the video
waxen hawk
His innocent look
waxen hawk
you should learn about mod-arg form and de moivres theorem before you can fully ...
I understand them
Ahhh
I learned about de moivre theorem
mellow oyster
well if you understand that complex numbers can be represented in polar form using e, then the nth roots of a complex number correspond to complex numbers where the angle of the polar form is divided into n parts, forming an n-gon in the complex plane
mellow oyster
bro are you asking a legitimate question or just messing around
waxen hawk
What’s the e stands for
I’m asking a legitimate question
I just so done with the e, that makes the whole situation becoming complicated
It is unnecessary
To express de moivre theorem using e
urban edge
the use of e is to take advantage of series to make the notation easier
waxen hawk
the use of e is to take advantage of series to make the notation easier
But what it has anything to do with Euler number
Is the e Euler number
Is the e Euler number
Or just a notation. A abbreviation for the whole shit
urban edge
just a notation trick
waxen hawk
For the whole de moivre that written in the form of trig function
urban edge
it is
waxen hawk
What
Wut
Why
Why
Then Is theee any benefit to use the notation e to express de moivre theorem
urban edge
note that youre not just using e, youre using e^something
some when you multiply things together, you can use exponent laws to make things easier
waxen hawk
I can still live without it right
Without knowing the e
The properties of it
I can still express my shit using de moivre
Without the e
urban edge
sure, but it makes life harder
waxen hawk
And it is totally fine
Where can I learn about e
It seems everything is correlated with e
And it has so many properties
That I’m unable to learn them all
urban edge
honestly im not sure
vast shale
Where can I learn about e
2.718281828…
waxen hawk
I didn’t know e has anything to do with de moivre theorem until I saw this video
urban edge
they come up as you go along, just make sure you understand them as best you can so when you learn a new property its easier
mellow oyster
I didn’t know e has anything to do with de moivre theorem until I saw this video
then study representations of complex numbers in polar form before continuing in the video
vast shale
But to be serious you can see the proof that e^itheta = cis theta if you’re interested
waxen hawk
then study representations of complex numbers in polar form before continuing in...
I did
mellow oyster
..
waxen hawk
mellow oyster
is your problem that you don't like representing cos(theta) + i*sin(theta) as e^(i*theta)?
waxen hawk
He’s so adorable
waxen hawk
is your problem that you don't like representing cos(theta) + i*sin(theta) as e^...
Yes
mellow oyster
that is absolutely standard when working with complex numbers and definitely useful to get used to
waxen hawk
is your problem that you don't like representing cos(theta) + i*sin(theta) as e^...
As I cannot see any benefit except sparing a few efforts from writing
mellow oyster
e isn't just a shorthand for this..
atomic forum
The exponential function just makes lots of things convenient
mellow oyster
e comes up in calculus, differential equations
basically everywhere
get used to it
atomic forum
e itself isn't that special
i mean the number does come up
but really it's the exponential function which is super useful
waxen hawk
but really it's the exponential function which is super useful
Why it is useful
waxen hawk
That’s fantastic
mellow oyster
Why it is useful
comes up all the time in differential equations because of its property of being its own derivative
atomic forum
the inverse, logarithm, turns multiplication into addition
waxen hawk
That’s marvelous
atomic forum
so it lets you convert between the two fundamental operations of arithmetic
mellow oyster
i feel like bro's being sarcastic 😭
atomic forum
addition and multiplication
waxen hawk
i feel like bro's being sarcastic 😭
No
atomic forum
i think that's pretty cool
waxen hawk
Im speaking the words of heart
waxen hawk
Im speaking the words of heart
What a strange expression
atomic forum
probably one of the best definitions of the exponential function is the power series one
$\exp(x) = \sum_{n=0}^\infty \frac{x^n}{n!}$
twin meteorBOT
Pseudonium
mellow oyster
and this lets you exponentiate some wild stuff that you might not think you'd be able to exponentiate
like matrices
or other operators
atomic forum
mhm
twin meteorBOT
desert hornet
**Pseudonium**
If you set x=i*theta and compare the imaginary and real parts you’ll get the series for sin(theta) and cos(theta)
atomic forum
yes
indeed this is how a lot of people define sin and cos
at least in modern math
twin meteorBOT
brittle cipher
Why it is useful
you can derive angle sum identities easily, they're really just exponent rules (which are easy ofc)
expand each of these as cis and compare real and imaginary parts: $e^{i(x+y)} = e^{ix} e^{iy}$
twin meteorBOT
Mero
brittle cipher
well idk about all that, but it can make things done in polar coordinates or 2D geometry kind of fun
another tool in your tool belt
like anything you learn about rotational motion in physics, try redoing them with the complex exponential, like the relations between position, velocity and acceleration for uniform circular motion as a specific example
vocal sleetBOT
@waxen hawk Has your question been resolved?
rough zenith
vocal sleetBOT
rough zenith
how did the area go from e^k/2 - 1/2 - k/2 to k/2 ?????
if you need to see hte original questiotn i can sent it too
vast shale
it should be this
Since the function passes through (k,k) substitute it in the given function
assuming it passes through (k,k)
so you are left with
area = k-k/2 = k/2
@rough zenith
vocal sleetBOT
@rough zenith Has your question been resolved?
vast shale
i dont think u can just convert it into that in an inequality
livid horizon
Are you sure it's (-1,0)?
atomic forum
$\log_{-x}$ looks very cursed
twin meteorBOT
Pseudonium
vast shale
fr
livid horizon
Fr
atomic forum
my first thought would be to use the base change formula
$\log_{-x}(a) = \frac{\ln(a)}{\ln(-x)}$
twin meteorBOT
Pseudonium
livid horizon
Wait if the base is less than 1 then the inequality sign changes ig
So you've to divide it into 2 cases, one where -x>1 and another where 0<-x<1
The first case has no solutions as you've already found
vocal sleetBOT
@sand ember Has your question been resolved?
signal pendant
Also, making the substitution -x=u will help clear up the wierd looking stuff
It looks a lot easier to handle, thats it.you want to solve $\frac{ln(u^2+4u+3)}{ln(u)} > 1$
twin meteorBOT
quickdoom
signal pendant
Make two cases, as stated above by sulfur
One where ln(u) is positive (1<u) and one where ln(u) is negative (0<u<1).
In the first case you will be able to multiply both sides by ln(u), but in the second case you will have to flip the sign of the inequality, since you will be multiplying by a negative number.
vocal sleetBOT
@sand ember Has your question been resolved?
rough zenith
vocal sleetBOT
rough zenith
no idea how to start
vast shale
Integrate :))
vast shale
a= dv/dt
vast shale
i dont really understand questino
what exactly do you not understand ?
vast shale
don't forget the c when you integrate
vocal sleetBOT
vocal sleetBOT
Available help channel!
twin rampart
vocal sleetBOT
twin rampart
How to find the inverse of the function above?
twin meteorBOT
ƒ(Why am. I here)=I don't Know
scenic ravine
for $x\neq -1$
twin meteorBOT
ƒ(Why am. I here)=I don't Know
scenic ravine
not yet
scenic ravine
atleast that's not how I'd do it
twin meteorBOT
neon
viral copper
It's because we're using the property of inverse functions:
[ f \circ f^{-1} (x) = f^{-1} \circ f (x) = x ]
viral copper
Now you can solve for the inverse function
rugged orchid
@viral copper ha
viral copper
go away
rugged orchid
Use \mapsto!!
viral copper
😭
twin rampart
we're replacing x with the inverse
But we haven't found the inverse yet, how can we replace?
rugged orchid
This is the not so scuffed way of doing “swap the places of x and y”
viral copper
Or are you questioning if there *exists* an inverse?
(which is valid, we only do this when we know that there exists an inverse)
twin rampart
Or are you questioning if there *exists* an inverse?
Hold on
In my lecture they taught what's an inverse function:
Let's say f: A -> B is one-to-one function (where B = R(f)). The function f^-1 : B -> A, f^-1 (y) = x is called the inverse of f: A -> B iff f(x) = y.
How do we find if the function is one-to-one?
viral copper
ughghghg one way inverses
viral copper
How do we find if the function is one-to-one?
Well by definition, if $x_1 \neq x_2 \implies f(x_1) \neq f(x_2)$, then the $f$ is one-to-one
twin meteorBOT
neon
viral copper
Or injective if you're bougie
twin rampart
Well by definition, if $x_1 \neq x_2 \implies f(x_1) \neq f(x_2)$, then the $f$ ...
And how do I find if some function is one-to-one or not?
cobalt crypt
check if it satisfies the definition
twin rampart
I think I'll not be checking each domain value
cobalt crypt
you must
cobalt crypt
not if you're smart about it
viral copper
It's probably easier to do with the contrapositive actually
cobalt crypt
id say your version is the contrapositive 
viral copper
[ f(x_1) = f(x_2) \implies x_1 = x_2 ]
twin meteorBOT
neon
cobalt crypt
this is the definition of injective thats usually presented
viral copper
id say your version is the contrapositive <:kekehands:1130985995102322719>
well
I was given the != one
rugged orchid
For all!!!
rugged orchid
I was given the != one
Wikipedia says otherwise
viral copper
Yeah wikpedia also uses the != one
twin rampart
not if you're smart about it
What do you mean?
cobalt crypt
What do you mean?
you can check for example that f(x) = x is injective
if f(a) = f(b), then a = b
so f is injective
twin rampart
you can check for example that f(x) = x is injective
what's injective?
twin rampart
Oh boy
twin rampart
if f(a) = f(b), then a = b
But how do I check that?
That should work for all the x value
cobalt crypt
f(a) = f(b) is the same as a = b by the definition of f that i gave
viral copper
Throw it in and simplify
cobalt crypt
f(x) = x
viral copper
horizontal line test?
twin rampart
For light functions - yes
twin rampart
also are we still on track for finding the inverse or?
Trying to understand how to find if function is one-to-one or not
twin rampart
Well, I still don't get if function is one-to-one or not
viral copper
why not
you started with f(x_1) = f(x_2)
and you've shown that necessarily x_1 = x_2
this is what it means for a function to be injective
twin rampart
But if function is one-to-one it should be x_1 != x_2
viral copper
no uh
twin rampart
twin meteorBOT
neon
twin rampart
It's the definition of one-to-one function from my lecture
viral copper
@cobalt crypt @rugged orchid take that
viral copper
It's the definition of one-to-one function from my lecture
yes you have the first definition
equivalent is the second definition
rugged orchid
<@118663865264242688> <@244390699431100416> take that
It’s not in English it doesn’t count
viral copper
you used the second one here
cobalt crypt
yes
rugged orchid
A => B is equivalent to not B => not A
rugged orchid
Draw out a truth table if you don’t believe it
It truth will reveal itself
Haha I’m so funny
twin rampart
So that's one-to-one function
viral copper
yes
twin rampart
Then why the inverse to the function is the same as the function?
viral copper
It's a result we notice
You can verify it
And if you attempt to find the inverse you'll see I was right
cobalt crypt
,w {{-1, 1}, {1, 1}}^-1
twin meteorBOT
viral copper
Assuming you don't need to show your work
Or stuff like that
But try to find the inverse normally first
twin rampart
But try to find the inverse normally first
Look what I get
viral copper
Let's hope it aligns with expectations
twin rampart
twin rampart
Let's hope it aligns with expectations
It surely doesn't
cobalt crypt
you still have a bit to go
twin rampart
you still have a bit to go
I agree
viral copper
also is that meant to be a 1
cobalt crypt
looks like a zeta
twin rampart
also is that meant to be a 1
Yep
cobalt crypt
,, x = \f {\zeta - y} {\zeta + y}
twin meteorBOT
twin rampart
I don't get what I'm supposed to do next
viral copper
isolate y
twin rampart
y = -x -xy +1
viral copper
Al'Jabr well help you
twin rampart
I'm probably the most unteachable guy
twin rampart
y = -x -xy +1
Man, there's -(x * y), how am I supposed to do that?
magic wren
Move it to the other side
twin rampart
Oh damn
twin rampart
Did you do it
Yep, it's the same as the original function
viral copper
Alright
twin rampart
Move it to the other side
Try B-major instead of those 4 notes
magic wren
Try B-major instead of those 4 notes
cobalt umbra
how to do this
vocal sleetBOT
how to do this
cobalt umbra
answer is C but how
fallen hawk
Opposite angles in a 4 sided figure are always equal to 180
fallow igloo
Opposite angles in a 4 sided figure are always equal to 180
not always
cobalt umbra
in circle right
fallow igloo
@cobalt umbra you know x= 108 since all angle are the same in a regular pentagon
cobalt umbra
yea that one is good
fallow igloo
since you know that you can find angle utp and vqp
and then you also have reflex p
now you get 2 equations
cobalt umbra
reflex p?
cobalt umbra
oh the obtuse outer one
cobalt umbra
Yes
fallow igloo
s and r are equal to 108
cobalt umbra
Yes
fallow igloo
so now from RSVU
you get angle V+U=180-108
which is 72
you get this part? @cobalt umbra
cobalt umbra
angle RVU + VUT?
cobalt umbra
not rlly
fallow igloo
we used angle sum property
since that is a 4 side fig its angle sum is 360
oh
i made little mistake
cobalt umbra
isnt it 5
fallow igloo
then you will get 2 equations for the angle
after that just solve like normal
and you should get y=72
cobalt umbra
ok
fallen hawk
I got it dw
The figure is kinda bad
Why is there a red dot on the rightside
@cobalt umbra
fallow igloo
wait i got it
we make a line from p pallallel to TU intersecting VU
from there we get a trapezium
the angle PTU=180-x
and base angle of trapezium are equal
so y=180-x
after that we just add x and 180-x
and get 180
fallen hawk
and base angle of trapezium are equal
Only if its isosceles
fallow igloo
bru
cobalt umbra
it isnt equal
because the issue with the question is that RV can be extendable
and if it extends than angle y can vary
fallow igloo
man idk
cobalt umbra
and theres no information regarding the length of RV or VU
cobalt umbra
dm me the answer pls
ok will do once someone figures it out
fallen hawk
I got it
cobalt umbra
so it makes right angle triangle V U and new line
i think its impossible
if the rv line can vary without affecting the pentagon etc and there is no information, then it is impossible without any length of RV
fallen hawk
What if we continue the line
TP
Continue it to VU
We'd know Q' is 72, we'd know P' is 72
We'd know that in total that would be 72+72+V-(180-72-Y)
vocal sleetBOT
@cobalt umbra Has your question been resolved?
cobalt umbra
if the rv line can vary without affecting the pentagon etc and there is no infor...
i am still adament this is the reason
vocal sleetBOT
@cobalt umbra Has your question been resolved?
solar barn
I also think it is impossible, that's only the case when VU is parallel to RS, which is not given in the problem.
solar barn
or alternatively, if TU = QV, which would lead to triangles RSW and WUV in this image to be similar, giving angle y=72, but neither this is given.
fallen hawk
If we draw line QT here
We'd get an isosceles triangle
And trapezoid QTUV
With 2 base angles equal to 108
no?
Ig we cant prove QT parallel to UV
vocal sleetBOT
@cobalt umbra Has your question been resolved?
harsh heart
Hey, my question is pretty simple, I have an inequalities like -4<x<3 and I want to surround x². How I can do that?
vocal sleetBOT
Hey, my question is pretty simple, I have an inequalities like -4<x<3 and I want...
harsh heart
And on ]-infinity ; 0[ x² is decreasing so we reverse the symbols
and finally 0<x²<9, right?
harsh heart
Ooh okay
hybrid flicker
So you'll end up choosing the inequalities with LESS constraints
harsh heart
So it's 0<= x² < 9 ?
harsh heart
0<x²<16
hybrid flicker
Yes
Coupled with 0<=x² < 9
We do agree 0<=x² is the less constraining for the LEFT inequality
What about the right inequality? Which one has LESS constraint?
harsh heart
For the right they have the same constraint no?
Oh no okay
I understand
It's 0<x²<16
hybrid flicker
Almossssst
harsh heart
Yes including 0
vocal sleetBOT
vocal sleetBOT
Available help channel!
harsh heart
Hey it's me again, I have an inequalities like -5<x<2 and I want to surround 1/x, how I can do that please?
vocal sleetBOT
Hey it's me again, I have an inequalities like -5<x<2 and I want to surround 1/x...
harsh heart
I have separated at 0 but of course we can't divide by 0 so I'm stuck here
Because I can't apply the function
magic wasp
If you want something like a < 1/x < b, you won't get it
magic wasp
Well because around 0 1/x goes from -inf to +inf
You can get something like 1/x < a OR 1/x > b
(or x = 0)
harsh heart
You can get something like `1/x < a OR 1/x > b`
And for my example what will be a and b?
magic wasp
Take -5 < x and x < 0 and make it 1/x < a
harsh heart
I got 1/x < -1/5
magic wasp
Yeah
harsh heart
And what do you do for x <0?
magic wasp
x < 0 is necessary to get 1/x < -1/5
magic wasp
?
harsh heart
Oh yeah
Okay so yes I won't write "and x<0" because it's useless
But what about x<2?
magic wasp
Yes
magic wasp
If it were an "and" you could write it 1/2 < 1/x < -1/5 which is just wrong
harsh heart
When we want to apply the square function and we have the same inequalities
We separate at 0
But why the " <= " is with the positive side?
Is that false to write <= on both side?
magic wasp
Example?
harsh heart
For example we have -5<x<2 and we want to surround x²
I separate the inequality at 0 and I got "-5<x<0 or 0<=x<2"
Why the "<=" is on the positive number side?
Is it wrong to write "-5<x<=0 or 0<=x<2"?
magic wasp
You can do -5<x<=0, it's just redundant
harsh heart
Okay
magic wasp
-5<=x is wrong though
harsh heart
Huh yeah I mistake something ^^
Okay and if we don't write any "<=" it's false
That's right?
magic wasp
Unlike in the 1/x case where 1/0 is undefined, here 0^2 is defined, so you can't just forget about 0
In the 1/x case you'd need to keep a "or x = 0" unless you really don't need it in the future
In the x^2 case you can just have x^2 = 0 as part of something like 0 <= x^2 < 25
It's the same as writing x^2 = 0 or 0 < x^2 < 25
magic wasp
harsh heart
.close
mellow geode
Consider a trigonometric series $a_0 + \sum_{k \geq 1} (a_k \cos(kt) + b_k \sin(kt))$.
- Prove that the following assertions are equivalent:
\begin{itemize}
\item[(i)] The numerical series $\sum_{k \geq 0} a_k$ and $\sum_{k \geq 1} b_k$ converge absolutely;
\item[(ii)] The trigonometric series is normally convergent over $\mathbb{R}$.
\end{itemize} - Prove that if conditions (i) and (ii) are satisfied, then the sum function $f : t \mapsto a_0 + \sum_{k=1}^{\infty} (a_k \cos(kt) + b_k \sin(kt))$ is continuous and $2\pi$-periodic.
vocal sleetBOT
Consider a trigonometric series $a_0 + \sum_{k \geq 1} (a_k \cos(kt) + b_k \sin(...
twin meteorBOT
abyssworld
hybrid flicker
!status
vocal sleetBOT
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
2
hybrid flicker
so what did you try?
mellow geode
we suppose sum of a_k and sum of b_k converge absolutely, so it means that sum of |a_k| and sum of |b_k| converge. We want to find a sup of the trogonometric series right?
mellow geode
i know that the sin(kt) and cos(kt) are bounded by 1
hybrid flicker
yep
mellow geode
so it means that the trigonomic function is:
ehm yes i found it thank you!!
latex is not working?
$\sum_{k \geq 1} \left|a_k \cos(kt)\right| + \left|b_k \sin(kt)\right| \leq \sum_{k \geq 1} \left|a_k\right| + \left|b_k\right|$
twin meteorBOT
abyssworld
mellow geode
.close
vocal sleetBOT
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last grotto
I want to graph the line,
y= (x - floored((x+1)/6)) +1
for values of x not divisible by 2,3
I also want to graph multiple other lines, and then see their valid outputs, where the lines all intersect.
last grotto
How would i go about doing this?
lilac plaza
What's floored?
last grotto
x+1 / 6
I'm basically looking for an online tool that will let me restrict the domain of inputs into a graph, or just give me the output after restricting the inputs by a divisibility
Its okay if there doesnt exist one, i can try to just code it
vocal sleetBOT
@last grotto Has your question been resolved?
true grail
What's floored?
rounded down to the nearst integer
looks like this
but that's over all x
last grotto
Yep! but i need the domain to not contain any inputs divisible by 2, or divisible by 3!
Exactly
last grotto
The problem is i need the intersections between many of these lines
true grail
well, u'd want to graph ur other lines and then see the intersections
last grotto
they are all like this
last grotto
all 5 lines i need to graph need to be domain limited
true grail
are you graphing by hand
last grotto
No, anything is fine i just need a list of these numbers that would be outputted
quickly that i can test
I just am verifying that my statements are true
true grail
in that case i'd just feed your answers into the function part of your caclulator and see if y changes
last grotto
i thought about it again, what would also suffice is just showing me all outputs that are shared by both functions.
sharp comet
vocal sleetBOT
sharp comet
I'm still confused with side splitter (triangle proportionality) theorem
and angle bisector theorem
so here I do 15/(x+15)
so the side of the small triangle divided by the side of the large triangle
=
8/12
so the same thing
15/x+15 = 8/12
cross multiplyy
8x+120=180
8x=60
x = 7.5
sharp comet
ok
shadow raft
but yeah, its correct
shadow raft
but angle bisector you dont use here
sharp comet
but angle bisector you dont use here
how do I know what to cross multiply
is it like x/4 = 9/6
or
6/4 = 9/x
idk
shadow raft
https://www.varsitytutors.com/hotmath/hotmath_help/topics/angle-bisector-theorem reading this might help
sharp comet
sharp comet
https://www.varsitytutors.com/hotmath/hotmath_help/topics/angle-bisector-theorem...
ok
how do I know what it says
for the image above
shadow raft
wdym?
sharp comet
like
how do I know what the angle bisector theorem says
a/b = c/d
I know that
but how do I implement that into my uestions on homework and tests
shadow raft
look at the bisector angle
sharp comet
so
sharp comet
if I look at the angle bisector
shadow raft
you can rewrite this to d/b=c/a
sharp comet
in this case should a be 4, b be x, c be 6, and d be 9
if I go by his video
where a and b are the bottom side
shadow raft
yep
shadow raft
it is
shadow raft
:D
shadow raft
they holding me back fr
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elder delta
i know this is a simple question but im kinda struggling
vocal sleetBOT
i know this is a simple question but im kinda struggling
shadow raft
can you rewrite tan as sin and cos?
vocal sleetBOT
@elder delta Has your question been resolved?
elder delta
do you mean as in tan = sin/cos in a unit circle?
vocal sleetBOT
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).
elder delta
i mean then the exact value of cos would be 15 no?
twin meteorBOT
Flappie
shadow raft
then write this as cos =?
elder delta
hmm
let me try...
hmm what do you mean is that not just 15 still? and does sin(thaeda) < 0 actually mean anything?
shadow raft
do you know how to cross multiply?
elder delta
yeah but why would that be necessary here?
elder delta
well i mean i just dont understand if sin/cos is equal to 5/15 wouldnt cos just be 15?
shadow raft
but sin is not 5?
elder delta
yeah thats true
shadow raft
\begin{align*}\frac{\sin}{\cos}&=\frac{5}{15}\15\sin&=5\cos\\cos&=\frac{15}{5}\sin\\cos&=3\sin<0\end{align*}
twin meteorBOT
Flappie
elder delta
ok well thanks a lot 👍 👍👍👍👍👍👍👍👍
vocal sleetBOT
@elder delta Has your question been resolved?
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Available help channel!
minor carbon
v is a constant vector
d/dt (v.v)