precalculus

[viscid thistle]

180+120

[slender river]

also why 180

[viscid thistle]

so 300

[slender river]

ull just go to the opposite side

and get 1/2 positive

ohec that's a smol one

[viscid thistle]

wait its -1/2 is that the same as 1/-2

[slender river]

yeah

[viscid thistle]

i thought it have to be exaclty the same

so long it equall the same

ok

so its 240

[slender river]

yes

[viscid thistle]

EASY

okay now

[slender river]

now u gotta do the general solutions

kek

[viscid thistle]

we right it as a infinite soultion

[slender river]

yus

[obsidian monolith]

ashiax:

[slender river]

late fren

lol

kek

i am rarted too kek

u need smort ppl around not me

like emeric

smh

[spring thunder]

TFW $\frac12$

[obsidian monolith]

emeric75:

[slender river]

lol

[viscid thistle]

so {x|x=120+-360n or 240+-360n, n eI}

did i do it right?

[spring thunder]

I gtg do the groceries tho

[slender river]

sure kuma

[viscid thistle]

wait is my soultion correct?

[spring thunder]

Speak broken like me

[viscid thistle]

even the eI part?

[slender river]

el

[viscid thistle]

lol o

[slender river]

ohec

do n \in N

or sth

$n \in \mathbb{N}$

[obsidian monolith]

MetalNinja27:

[viscid thistle]

i thought it was an i

wait so it would be {x|x=120 or x=240+-360n, n e i

i thought n e i means n has to be an interger

[slender river]

oyeah sure

negative integres i forgot

but u already had plus minus so

ehhh

@crisp grotto oi

wat u need help on

i hope it's something i know tho

i get like 18 million pings

send channel name

[viscid thistle]

wait

{x|x=120degree+-360degree n or x=240 degree +- 360 n, n e i}

why is there +-360?

if we did not even use 360

[slender river]

0 is an integre

[viscid thistle]

theta?

theta is an interer

[slender river]

wat

[languid crane]

what is integre

[slender river]

lol

[languid crane]

wan learn

[viscid thistle]

x is theta i was too lazy to type theta all the time

or do u mean n is an interger?

[slender river]

n integer so yeah

0 is an integer too

so itll return the correct solutions

[viscid thistle]

0 as in theta?

[slender river]

0 as in n

[viscid thistle]

oh ok

[slender river]

yeah

[viscid thistle]

find all the soultions for 6Sqrt2csc(theta)-4Sqrt6=0

[charred hull]

is it

$6\sqrt{2csc(\theta)} - 4\sqrt6=0$

[obsidian monolith]

Colorodo Brown Stain:

[charred hull]

@viscid thistle

[viscid thistle]

hmm well the Sqrt is only under 2

[charred hull]

$6\sqrt2\csc(\theta) - 4\sqrt6=0$

?

[willow bear]

\csc

[viscid thistle]

yea

[obsidian monolith]

Colorodo Brown Stain:

[viscid thistle]

looks like that

yea

[charred hull]

what have you tried so far

[viscid thistle]

like that

[charred hull]

look for the possible values of $\theta$

[viscid thistle]

i was thinking of trying to divide csc by 1 to get it as sin

[obsidian monolith]

Colorodo Brown Stain:

[willow bear]

isolate csc(θ) first!

[viscid thistle]

but i also want to get sin alone

but idk how do that with the sqrt

[charred hull]

where's sin

[viscid thistle]

well i mean csc

[willow bear]

isolate csc(θ), then take the reciprocal of both sides

[viscid thistle]

i thought i could solve for it as sin

oh ok

so can we add add -4Sqrt6 to both sides?

[charred hull]

yes

[viscid thistle]

ah okay so then 6sqrt2csc(x)=4Sqrt6

[charred hull]

wait

[willow bear]

no

[charred hull]

ah typo

[viscid thistle]

yea mb

[charred hull]

where's the - from

[viscid thistle]

oops

[willow bear]

you wouldn't go from 7x - 11 = 0 to 7x = -11

[viscid thistle]

yea mb

6sqrt2csc(x)=4Sqrt6

[charred hull]

add 4sqrt6

ok

[viscid thistle]

so this is the correct

so now we can simplify csc to sin?

[willow bear]

no not yet, you've yet to isolate it

[charred hull]

isolate csc completely

[willow bear]

i mean you can but

it's a bit of a hassle

[viscid thistle]

wait how can we isolate it if 6Sqrt2 is part of csc

[willow bear]

6sqrt(2) is multiplying it.

it's not "part" of the csc.

[charred hull]

cancelling out the multiplication

[viscid thistle]

oh i see

[willow bear]

anyway i'm off to sleep i'll let colo handle the rest of this

[viscid thistle]

okay

[charred hull]

night ann

[viscid thistle]

so we simplify 6Sqrt2

so it would be 8.48?

[charred hull]

wat

if you want to turn 6sqrt2 into 1

what do you do

[viscid thistle]

hmm would we subtract 6 to both sides

[charred hull]

no

if you had an equation $3x=3$

what would you do

[viscid thistle]

oh we would make it as 6*2^2

[obsidian monolith]

Colorodo Brown Stain:

[viscid thistle]

oh

divide 3 by both sids

[charred hull]

yeah

[viscid thistle]

so you would just divide it by it self then

ah okay

[charred hull]

yeah

so divide both sides with?

[viscid thistle]

but i always thought you would have to do that to both sides right?

[charred hull]

correct

you can divide the other side still

[viscid thistle]

oh so we divide 6sqrt2csc(x)=4Sqrt6 both sides by 6Sqrt2

[charred hull]

yes

just write the right side as a fraction

[viscid thistle]

ah okay so then we have 1csc(x)=4/6Sqrt2/2 2/2=1

[charred hull]

that

is not the cas

*case

[viscid thistle]

hmm

[charred hull]

$\frac{4\sqrt6}{6\sqrt2}$

gdi

[obsidian monolith]

Colorodo Brown Stain:

[viscid thistle]

OH

okay

[charred hull]

so dont simplify it yet

[viscid thistle]

ah ok

[charred hull]

now we have csc equal to that right

and since csc is 1\sin

you want to isolate sine

so you would take the reciprocal of both sides

[viscid thistle]

wait

it would be 1*csc(x)

then it would be csc(x)

right?

[charred hull]

yeah

[viscid thistle]

okay now its alone we turn it to 1/sin

which then turns to sin(x)

[charred hull]

wait

doesnt just turn into it, you take reciprocal both sides

so what does the other side turn into?

[viscid thistle]

wait why do we have to take the reciporcal of both sides

the other side does not have sin or csc

[charred hull]

1/sin does not equal to sin

[viscid thistle]

oh yea

[charred hull]

you want to flip one side over, you have to flip the other

[viscid thistle]

wait why couldnt we jus thave divided it by 1

[charred hull]

dividing something by 1 doesnt change it's value

[viscid thistle]

no like 1/csc=sin

[charred hull]

1/3 divided by 1 doesnt turn into 3

[viscid thistle]

since now we got csc alone

if we divide csc by 1

it would turn to sin then because thats its reciporcal

[charred hull]

that doesnt turn into 1

if you divide 1/3 by 1 it doesnt turn into 3/1

[viscid thistle]

oh ok

[charred hull]

you would divide 1 by (1/3)

[viscid thistle]

so now we turned csc to 1/sin

so we so now we have 1/sin(x)=4Sqrt6/6Sqrt2

[charred hull]

yes

now: reciprocals

[viscid thistle]

hmmm?

[charred hull]

you want to find the reciprocals

to go from 1/sin to sin

[viscid thistle]

ah yes

but how?

[charred hull]

reciprocal

reciprocal of 1/x is x

[viscid thistle]

so it just changes to from 1/sin to sin

[charred hull]

do that to both sides

[viscid thistle]

so i would also flip it to 6Sqrt2/4Sqrt6

[charred hull]

yes

[viscid thistle]

ah because we flipped 1/sin to sin/1 so we had to do that to the other side as well

and sin/1=sin

[charred hull]

yes

[viscid thistle]

ah okay

so now sin=6Sqrt2/4sqrt6

[charred hull]

what you do to one side, you do to the other

yep

[viscid thistle]

so now we can simplify 6Sqrt2/4sqrt6

[charred hull]

i dunno if you have to, but might as well simplify the right side right

[viscid thistle]

yea

so sin(x)=8.48/9.79

so sin(x)0.86

[charred hull]

that's not exactly simplification

[viscid thistle]

was i supposed to take the inverse?

sin inverse

[charred hull]

i was thinking you get it into the smallest fraction form you could

but yeah use inv. sine aka arcsin

[viscid thistle]

so sin^-1(0.86)=60

[charred hull]

yep

although

[viscid thistle]

ah

[charred hull]

you dont need a calculator for that

[viscid thistle]

sin is y

[charred hull]

yeah if you simplified it in fraction form

[viscid thistle]

so 60 is in quadrants 1 and since 0.86 is a positive it would be in quadrants 2

[charred hull]

you would have gotten sqrt3/2

[viscid thistle]

so 180-60=120

how would you get sqrt3/2?

[charred hull]

6sqrt2/4sqrt6

one sec

[echo plaza]

latex it

[charred hull]

yeah im on my phone

it's semi tough to kek

$\frac{6\sqrt2}{4\sqrt6}=\frac{3\sqrt2}{2\sqrt6}=\frac{3\sqrt2}{2\sqrt2\sqrt3}=\frac{3}{2\sqrt3}=\frac{\sqrt3}{2}$

[echo plaza]

[viscid thistle]

thats still confuses me can you describe each of the steps?

[echo plaza]

[charred hull]

its not loading kek

oof

[obsidian monolith]

Colorodo Brown Stain:

[charred hull]

do you sorta get it?

[viscid thistle]

i don't think it matters since Sqrt3/2 is the same on the unit circle as 120 while Sin^-1(0.86)=120

[charred hull]

theta in this case can be 60 or 120

[echo plaza]

u wot

[viscid thistle]

to be precise 0.86618999

[charred hull]

furthermore simplifying like this means you can use unit circle or 30/60/90 + sohcahtoa to figure this out

[viscid thistle]

you know ill just go with your method

your method is more accurate XD

[charred hull]

idk if youll ever not be able to use ur calculator but it does guarrantee you can solve this without it

[viscid thistle]

so i can just say then 6Sqrt2/4Sqrt6=Sqrt3/2 then?

[charred hull]

yeah after simplifying it

[viscid thistle]

so then Sqrt3/2=120

[echo plaza]

$\sin(120^{\circ})=\frac{\sqrt{3}}{2}$

[obsidian monolith]

CaptainLightning:

[echo plaza]

sqrt(3)/2 is the sin of it

[viscid thistle]

oh so then Sin^-1(Sqrt(3)/2

=120

[charred hull]

you can get theta to be: 60 + 360n or 120 + 360n

try sin(60) first

[viscid thistle]

hmm?

wait 60 is a degree

its not a radian

sin(60) gave me 0.8660

[charred hull]

yeah

sin 60 = sin 120

[viscid thistle]

yea

[echo plaza]

$120^{\circ}$ in radians is like $\frac{2\pi}{3}$ or something

[obsidian monolith]

CaptainLightning:

[echo plaza]

if that's what you're looking for

[viscid thistle]

so can i just say that sin^-1(Sqrt(3)/2=120

[echo plaza]

well

it would be 60 degrees not 120

[viscid thistle]

yea it gave me 60

[charred hull]

find all the soultions for 6Sqrt2csc(theta)-4Sqrt6=0

[viscid thistle]

yea

[charred hull]

you need all possible solutions

[echo plaza]

because arcsin is between -90 degrees and 90 degrees

[viscid thistle]

oh so then Sqrt3/2 is just 120

[charred hull]

so: 60, 120, and the cycles around need to be listed

[viscid thistle]

based off the unit circle

[echo plaza]

if you're solving an equation you have to look at the rest

[viscid thistle]

right we get 120 from the unit circle because Sqrt3/2 is on the second quadrant and it gives us 120

[echo plaza]

what's the interval you're supposed to solve it in?

[viscid thistle]

hmm?

the range is sin right?

and sin is positive

so that would mean its in the second quad

[echo plaza]

there

[charred hull]

why are you so focused on 120 degrees

[echo plaza]

do you want all of them?

[charred hull]

it also includes 1st quadrant

[viscid thistle]

there is no interval

yea its infinite

[echo plaza]

ok

[viscid thistle]

yea and 60 degrees is in the first quad

so now we right the soultion as infinitie

[echo plaza]

no

[charred hull]

where did infinite come from

[echo plaza]

you write down for which x it works

there are infinitely many but you need to say which

[viscid thistle]

so {x|x=60+-360degreen or x=120degree+-360degreesn, n e i}

[echo plaza]

ye

[viscid thistle]

it has no fixed interval it wants all soultions so it would be infinite soultions right?

[charred hull]

yes

[viscid thistle]

so i would right it as the soultion {x|x=60+-360degreen or x=120degree+-360degreesn, n e i}

[echo plaza]

you can just say + instead of +-

n can be negative

[viscid thistle]

is it sitill correct to right it as {x|x=60+-360degreen or x=120degree+-360degreesn, n e i} ?

[charred hull]

• is not necessary

it results in double counting your answers which can get points docked off

[echo plaza]

also the symbol for the set of integers is $\mathbb{Z}$

[obsidian monolith]

CaptainLightning:

[echo plaza]

not i

[charred hull]

$n\in\mathbb{Z}$

[obsidian monolith]

Colorodo Brown Stain:

[viscid thistle]

is it z because of this specically

or is this in general?

[charred hull]

Z means integers

[echo plaza]

in general

[viscid thistle]

thats wierd

[spring thunder]

Z means zahlen

[charred hull]

kek

[echo plaza]

sounds german

[spring thunder]

it is

[echo plaza]

roll with it

[charred hull]

represents integers

[viscid thistle]

i was never taught that they only teached me it as i even in the books

[echo plaza]

oh then just do i I guess

[viscid thistle]

and i never saw them right n as a negative in the books

it was always +-

[charred hull]

what

[viscid thistle]

yea like never

[echo plaza]

maybe the book writers don't believe that negative numbers can be integers

[charred hull]

just roll with that then

[viscid thistle]

no lol

[charred hull]

probably cause it's easier to explain

[viscid thistle]

yea

aite thanks guys

[charred hull]

np

[viscid thistle]

can anyone check my work

find all soultions 3cos(3θ) – 8 = –11 on the interval of [0°, 360°)

So we first add 8 to both sides this then gives us 3cos(3theta)=-3 we then divide -3 by 3 then we have 3theta=cos^-1(-1) since cosine is a negative it will be in quadrants 2 and 3 we then simplified the equation to 3theta=180 degree then we divide 3 to both sides and get theta=60 degrees since cosine is in quadrants 2 and 3 we do 180+60=240 and 180-60=120 so our angles are {60 degrees, 120 degrees, 180 degrees}

is this correct?

<@&286206848099549185>

[hoary yoke]

uh so imagine it like this

let 3theta = x

so boundry will be [0;120)

[viscid thistle]

wait its wrong?

[hoary yoke]

cant say its wrong

[viscid thistle]

oh ok

so 3theta=theta?

[hoary yoke]

wait no

[viscid thistle]

oh ok x

[hoary yoke]

uh how can i say

if you solve the equation you are given

[viscid thistle]

hmm..

[hoary yoke]

you should get the answer like

theta = (360n-180)/3

where n is a whole number

only solution i can think of which suits your equation is 60 degrees tbh

[viscid thistle]

yeah, yo

[hoary yoke]

wait lemme simplify that

theta = 120n-60

[obsidian monolith]

Chris, who does math:

[hoary yoke]

this is in degrees

not radians

pi cant be used

[viscid thistle]

wait

so the two soultions are 60 then?

[hoary yoke]

i messed up

so plug in 0 1 and 2 as n in the formula i gave

theta = 120n-60

theta = -60 (wont work)

theta = 60 (works)

theta = 180 (works)

[viscid thistle]

u can look at graph of cos(3x)=-1

[hoary yoke]

theta = 300 shud also work by that logic tbh

60;180;300 shud be the solutions i think

[viscid thistle]

are they all the same as 60,180,300?

[hoary yoke]

you need to test that tbh

[viscid thistle]

how do you test it

[hoary yoke]

plug in the value

[viscid thistle]

like is there a way to check your answer?

how do i do that

which is 60 degrees, 180, and 300

so yeah boii u right

im doing analysis hw, i should pay better attention

wait how do you get 180 and 300?

just graph that shit my dude

desmos, then put in cos(3x)=-1, then look at graph from 0 to 2pi

[hoary yoke]

all 3 solutions work

[viscid thistle]

which is the same as 0 degrees

to 360

no i mean how do you find the soultions 180 and 300

[hoary yoke]

Kuma so first what you shouldve done is solved for cos(x) solutions repetition

which is

cos(x)=360n-180

it repeats the same value

for every whole number "n"

[viscid thistle]

wait

what you mean by cos(x) repetion?

[hoary yoke]

it repeats same solution every 360n-180 degrees

lol

[viscid thistle]

what

[hoary yoke]

uh let me demonstrate

so you have for example

-1

as your solution for cos(x)

[viscid thistle]

3cos(3θ) – 8 = –11 should'nt you first add 8 to both sides of this equation?

[hoary yoke]

ye

you solved that process

[viscid thistle]

so everything i did to get 60 was correct?

[hoary yoke]

yep

[viscid thistle]

so it was after that which was wrong

[hoary yoke]

ye

i mean some solutions matched

[viscid thistle]

yea only 120 was wrong then?

[hoary yoke]

mhm

dud i forgot this topic hard

😦 i feel bad

[viscid thistle]

is there someone else online who is know this more good?

[hoary yoke]

-1 is only for angle 180 right?

that is cos(x) = -1 only for x=180

right?

[viscid thistle]

confusing

[hoary yoke]

wdym confusing

im saying that

cos(180)=-1 right?

[viscid thistle]

oh yes

[hoary yoke]

so cos is -1 only on angle 180

but actually

[viscid thistle]

wait

you mean 3cos(x)=-1

[hoary yoke]

nooo

i mean normal cos(x)=-1

[viscid thistle]

ok

[hoary yoke]

when you solve it normally you will get answer 180 degrees

but thats not all the solution

if from 180 degrees, circle makes another full rotation (360) degrees

it will still end up in the same spot right?

since he is going on a circle

[viscid thistle]

then i divided 180 by 3 and got 60

[hoary yoke]

no no

you are rushin it

let me explain

[viscid thistle]

but thats what i did

is that wrong?

[hoary yoke]

kind of kind of not

there are more efficient ways to solve it

[viscid thistle]

oh i see

[hoary yoke]

and im showing you one of that way

[viscid thistle]

so 180 itself is also a degree

[hoary yoke]

yep

so if you stay at 180 degree

[viscid thistle]

so thats a soultion already okay

[hoary yoke]

and make a full turn (360 degrees)

will you still end up in a same spot?

[viscid thistle]

yes

no

[hoary yoke]

o.o

[viscid thistle]

you would be at 360+180

[hoary yoke]

yep

cos(180)=cos(360+180)

[viscid thistle]

ah i see

[hoary yoke]

360 is like adding 0

[viscid thistle]

yea

[hoary yoke]

bcs it doesnt change a value

[viscid thistle]

yes

[hoary yoke]

so the function cos(x)=-1 repeats its values EVERY 360 degrees + 180

every 360 degree can be written as 360n

where n is a rotation amount

so cos(x)=-1

x=360n+180

for our case x is 3Theta

so

3Theta = 360n+180

Theta = 120n+60

now what you can do is

[viscid thistle]

oh i see

so i forgot to add the 360n

[hoary yoke]

for formula 120n+60 find n so that 120n+60 is on the interval [0;360)

and when you find the solutions you will get 3 of them

one being 60

another being 180

and 3rd being 300

i tested all of them and all of them work

[viscid thistle]

how did you get 180 and 300? and 60

[hoary yoke]

ok so 120n+60

[viscid thistle]

yea

[hoary yoke]

lets try plugging in 0 and see what we get

e get 60

60 is on the interval [0;360) right?

what this is basically asking you is

is 60 a number when counting from 0 to 360

[viscid thistle]

wait

why you plug zero in n

[hoary yoke]

just for testing

[viscid thistle]

no i don't mean for testing but like how did you find 60, 180, 300

[hoary yoke]

im explaining that rn

[viscid thistle]

k

[hoary yoke]

i need your confirmation

does 60 lie on the interval [0;360)

[viscid thistle]

yes

[hoary yoke]

ok so thats the first solution

hmm lets now test n=1

[viscid thistle]

but how did you get 60?

[hoary yoke]

120(0)+60 = 0+60=60

functions

[viscid thistle]

thats how you solved for 60?

[hoary yoke]

yep

selected n randomly

and saw if it was on interval

if it is then its a solution

since 120n+60 is a solution to theta

but you are bounded

most people mess up Boundry and its rly hard to keep track

i would show you the full solution

but i think youd get confused

[viscid thistle]

show me full solution

[hoary yoke]

ok so we know the boundry goes from 0 to 360

[viscid thistle]

hey

[hoary yoke]

so we will see where function theta(n)=120n+60's n value goes from

[viscid thistle]

360-60=300

[hoary yoke]

meaning we make it equal to boundry

120n+60=0 => n=-1/2 which cant be since n is restricted to be only whole numbers

[viscid thistle]

wait

[hoary yoke]

thus meaning lower bound of n is 0

[viscid thistle]

cosine is x right?

[hoary yoke]

i just got a random value to demonstrate

cos(x)=-1 => x=360n+180

but for your case x was 3 times as big

meaning solution was 120n+60

divided everything by 3

ill continue

[viscid thistle]

aint cosine in the fourth quaddrants?

so wouldnt 360-60=300

[hoary yoke]

cosine is in all quadrants

[viscid thistle]

no like astc

[hoary yoke]

yes that solution is true

but it confuses me

tbh you shud go with solution you know

i solved it my way

[viscid thistle]

wait so

so 180/3=60

[hoary yoke]

60 degrees is the MAIN solution

[viscid thistle]

and we got 180 from the cos^-1(-1)

[hoary yoke]

yep

[viscid thistle]

so thats alrdy 2 soultions

and if cosine is in quads 4

[hoary yoke]

idk cosine being in 4th quadrant bs tbh

[viscid thistle]

because of astc

woul'dnt that man 360-60=300

cosine is negative right and it is x?

[hoary yoke]

.-.

cosine is positive

so yes

[viscid thistle]

like cosine is domain

[hoary yoke]

it would

[viscid thistle]

no cosine is negative

remmber it was -3/3

[hoary yoke]

cosine is POSITIVE in 4th quadrant isnt it

o.o

[viscid thistle]

no i mean in our soultion

is the cosine in our soultion a negative or a positive

its a negative right?

[hoary yoke]

oh no you cant solve it that easily LOL

300 there would mean 100 in actual solution

thats incorrect

you should check where 900 is

[viscid thistle]

no i said 360-60

[hoary yoke]

lol

[viscid thistle]

because the fourth quad states

360-theta=

so 360-60

[hoary yoke]

Kuma

[viscid thistle]

yes

[hoary yoke]

you dont understand something i think

ok

so

you have 3THETA as an angle

[viscid thistle]

yea its 180

[hoary yoke]

and you have only THETA restricted

NOT 3THETA

so 3THETA restriction would be this

[viscid thistle]

3theta=180

and when u divide it by 3

you get theta=60

so 360-60

gets 300

[hoary yoke]

im gona cri

[viscid thistle]

nooo

[hoary yoke]

3Theta restriction is 0;1080

you should find angles there

but you wont be happy finding it there bcs its hard

[viscid thistle]

then i found 60

[hoary yoke]

3 theta has to be same as 180

not EQUAL to

Theta can be 60 bcs 3*60 is 180

180 ~ 180 in terms of solution

theta can be 180

bcs 3*180 = 360+180 ~ 180

bcs 360 is practically a 0

and finally 300 can be a solution bcs

[viscid thistle]

yea i agree 60, 180, and 300 are all soultions

[hoary yoke]

3x300 = 5x180 = 2*360+180 ~ 180

AGAIN

360 means practically nothing

[viscid thistle]

yea 360 like x?

[hoary yoke]

wdym like x?

[viscid thistle]

like its not a number

[hoary yoke]

360 degrees

cos(360) = 1

[viscid thistle]

oh so its 0

[hoary yoke]

oops

lol

[viscid thistle]

1

[hoary yoke]

but 360 rotation means nothing

if it rotates 360 it will end up in the same spot

[viscid thistle]

i did cos(360) in m caulator

and did not get 1

[hoary yoke]

bcs you have it in radians

[viscid thistle]

i had it in radians

[hoary yoke]

-0.28369109148

you got this didnt you

[viscid thistle]

yes

[hoary yoke]

ye you have it in radians

we are talking about degrees

[viscid thistle]

so if 60, 180, 300 are correct why is the way i found them wrong?

[hoary yoke]

bcs your 300 find is same as finding 100

bcs 3*100=300

[viscid thistle]

what about yours?

[hoary yoke]

we are talking about theta that will do 3 rotations around the circle

we have to find all solutions theta which will be same value (-1) on all of the occasions

[viscid thistle]

wait a minute

i can only use 360-60 if cosine is positive right?

and cosine is a negative here

[hoary yoke]

👏

clappers

u got it

[viscid thistle]

so what do i do now

how do i find 300 the correct way

[hoary yoke]

basically you know your solutions are in quadrant 2 and 3

since cosine is negative

[viscid thistle]

i found 180 through cos^-1(1) and got 180 then i divided 180 by 3 and got 60 so i have 2 soultions of 60 degrees, and 180 degrees but not 300 yet

yes

[hoary yoke]

you dont know how u got solution 180 first

lol

yu only know 60 degrees

[viscid thistle]

cos^-1(-1)

[hoary yoke]

that is not how u solve for solution 180

[viscid thistle]

but it gave me 180

[hoary yoke]

but you mean 3theta=180

not theta

ll

[viscid thistle]

o

so i olnly know 60

[hoary yoke]

yep

[viscid thistle]

how do i find 180

[hoary yoke]

add odd number of 180s to it

lol

[viscid thistle]

180+60

[hoary yoke]

you see

[viscid thistle]

240?

[hoary yoke]

180+180+180 = 180+360

as i told you 360 means nothing

it is just a rotation

[viscid thistle]

yea

[hoary yoke]

it will be same solution

so 180+360 which is 3*180 will get us same solution as 180

so we get that

3*180=180

so 180 is a solution

lets rotate 180 odd number times again

(we are given 3 rotations. use it)

[viscid thistle]

that's just testing tho to find 180

is there like a proper way to find 180

because i have to show my work

[hoary yoke]

i told you my proper way

wait i know someone who might help

its 3am here

[viscid thistle]

o

i understand your way is correct

[hoary yoke]

@spring thunder dud eme come here plz i forgot half of the trig

[viscid thistle]

but i was never taught that wya

[hoary yoke]

ye understandable

find all soultions 3cos(3θ) – 8 = –11 on the interval of [0°, 360°)

this is the question

he could find 1/3 solutions

[spring thunder]

yeah usually i don't do it with that changing the range thingy

so yeah cos(3t) = -1 lel

[viscid thistle]

yea

[spring thunder]

so i'm gonna do like if we solved the equation in mega general form

[viscid thistle]

do you then do cos^-1(-1)

[spring thunder]

(ie in R)

[hoary yoke]

x=360n+180 <- this cannot be used

[spring thunder]

yeah cos^(-1)(-1) = 180 degrees

[hoary yoke]

he told me he hasnt learnt it yet

i honestly donno

[spring thunder]

1.rip 2. 360n+180 is not even hard to grasp conceptually

[viscid thistle]

btw it is on the interval [0, 360)

[hoary yoke]

kuma i alrdy explained

theta is on the interval

[viscid thistle]

ok

[hoary yoke]

which is like

rly big thing to get

[viscid thistle]

no

i learned 180+360n

so

i did learn about x=360n+180

[spring thunder]

cos(3t) = -1 : well if we take inverse cos on both sides (taking care of the fact that if we add/substract 360 degrees we also get a solution)

we get $3t = 180 + 360n; ; n\in \mathbb{Z}$

[obsidian monolith]

emeric75:

[viscid thistle]

yea

[hoary yoke]

[viscid thistle]

then divide it all by 3

[spring thunder]

ie $t = 60 + 120n; ; n\in\mathbb{Z}$

[obsidian monolith]

emeric75:

[spring thunder]

now you just gotta find the values of n that make t in [0,360)

[hoary yoke]

exact same method

[spring thunder]

(by trial and error, or solving a little inequality for difficult cases)

yeah it is lel

[hoary yoke]

Spoilers n ranges from 0 to 2 by Restrictions for "n"

so only solutions are n=0;1;2

[viscid thistle]

so x=60+120n

this makes 60 a soultion right?

and when we add 60 and 180

i mean 120

[hoary yoke]

ye

[viscid thistle]

we get 180

[hoary yoke]

yep

[viscid thistle]

so thats also a soultion

[hoary yoke]

180+120 = 300

300<360

works

and must be the solution

[viscid thistle]

im a bit slow let my brain cells work

[hoary yoke]

but 300+120=420>300

which doesnt work

so only 3 solutions

6;180;300

[viscid thistle]

yea it cant be 420

[hoary yoke]

i tested and all 3 of them work as i said

😄

so youre gucci

[viscid thistle]

wait

so in order to find 300

[hoary yoke]

[viscid thistle]

180+60

240

then we

divide by 3?

[hoary yoke]

i just remembered how to find all of it

welp too late

lol

[viscid thistle]

how

can you tell me

[hoary yoke]

well

[viscid thistle]

how to find 300

[hoary yoke]

we know we have 3theta

[viscid thistle]

we already found 60 and 180

[hoary yoke]

which means we have 3 full rotations

[viscid thistle]

yes

[hoary yoke]

so 3x360=6x180

so we have 6 180 rotations

if 180 multiple is even we will get 0 as the answer since it will be a multiple of 360 => 2nx180=nx360

if 180 multiple is odd we will get -1

so we know

3theta = 1x180;3x180;5x180

solving for theta will give us

theta=60;180;300

lol

it just hit me

[slow wharf]

Any trick to solving

$\int \frac{(arcsinx)^2}{x^3}dx$

[obsidian monolith]

Autistic Hoodie:

[slow wharf]

It seems to be very long when I try to solve it

[willow bear]

i'd try the subtitution x := sin(u)

[slow wharf]

hmm

[willow bear]

also idk why you're posting this here when it clearly belongs in #calculus (or a questions channel)

[slow wharf]

Sorry, I thought this was all calculus

[languid crane]

Precalculus != calculus

[slow wharf]

It sounds similar :/

[languid crane]

pre tells u

[slow wharf]

derivatives? Imaginary numbers?

[languid crane]

like preschool is not equal to school

derivatives are calc

[serene heath]

just read channel topic tbh

[slow wharf]

Oh

Thank you

[distant flume]

" A box with square base and no top is to hold a volume 100. Find the dimensions of the box that requires the least material for the five sides. Also find the ratio of height to side of the base"

I don't understand what it's asking me to do

[willow bear]

you've got a box

and it's got two dimensions you can control

the base length, and the height

[echo plaza]

like this

[willow bear]

like that, but with a square base.

[distant flume]

I get that

[echo plaza]

except square base

[willow bear]

and you know that the volume of the box must be 100

and what you're asked to do is find the dimensions that make the area of the box as small as possible under that constraint

[distant flume]

I don't get why it asks me to calc what would use less material

Isn't it the same for every lenght we choose?

[echo plaza]

I wouldn't expect so

[distant flume]

For example a 10x10 base and 1 height, or a 1x1 base with 100 height

[echo plaza]

the 10x10 base and 1 height uses less doesn't it?

[willow bear]

10x10 base with height 1 uses 140 square units of material, while 1x1 base with height 100 uses 401 square units of material

[distant flume]

Does it? If you take the 10x10 one and put it on its side it's the same as the 100 one

[willow bear]

no it's not

the box is three-dimensional

[distant flume]

Wait, how does that work then?

[echo plaza]

you've got 4 of these long things

each with area 100

and 1 square with area 1

then the 10x10 base one is like

the square with area 100

and 4 long things each with area 10

[distant flume]

So is cube root of 100 the solution?

[willow bear]

how did you arrive at that?

[distant flume]

I used logics, after thinking about the two examples I gave before

[willow bear]

"used logics"?

[distant flume]

Yeah well it simply makes sense, as they approach each other the material approaches somewhere in the 90s

The 4.61x4.61x4.61 box seems legit

If I have a 5x5x4 box for example, the answer is 115, which is less than the other options I gave before

[willow bear]

as they approach each other
who

[distant flume]

As the dimensions approach each other

[echo plaza]

you can write the height in terms of the square side length btw

because you know the volume

[distant flume]

It's like solving a system where Base x Height=100 but we also have to find the minimum value for the sum of the base and height divided in its components

[echo plaza]

yes

that's what you want

so

find an expression for the surface area

in terms of the square side length

[distant flume]

Yeah so isn't it 4.61? approximatively?

[echo plaza]

it might be but it could easily not be

[distant flume]

100=(b^2)+(h*4)?

[willow bear]

but base * height != 100.

and b^2 + 4h is not 100 either.

the volume is $b^2h$

[obsidian monolith]

Ann:

[willow bear]

and the area of material is $b^2 + 4bh$

[obsidian monolith]

Ann:

[echo plaza]

I think he means the base square

when he says the base

[distant flume]

I mean b isn't base, it's the length of the base

Yeah

[willow bear]

still

bh is not 100

[distant flume]

It shouldn't be

[willow bear]

but you claimed it was.

[distant flume]

What do you mean exactly

[willow bear]

It's like solving a system where Base x Height=100

[distant flume]

Indeed

Where base=b^2

[willow bear]

you failed to make that clear

[distant flume]

I said it was the length of the base

[echo plaza]

I mean this might be a language thing but I probably wouldn't call the length of the square the base

I would call its area the base

[distant flume]

Alright

Then let's just say that 100= base x height

But since base=n^2 (n=number)

And height=4m (m=another number)

Oh wait

Well anyway, it's as I said before 100=n^2+4h

[willow bear]

no

[distant flume]

You're right I'm confused now

[willow bear]

you are, very much so

[distant flume]

I wouldn't be if you helped me man

[willow bear]

well, it was you who started presenting your attempt

okay so like

this is the most straightforward method to do it

let x and h be the base length & the height of the box

its volume is then expressed as $x^2h$, which we are given is equal to $100$, so we know $h = 100x^{-2}$

[obsidian monolith]

Ann:

[willow bear]

and the area of material used to make the box is expressed as $A = x^2 + 4hx$

[obsidian monolith]

Ann:

[distant flume]

I see

[willow bear]

which, given our constraint, can be written as $A(x) = x^2 + 400x^{-1}$

[obsidian monolith]

Ann:

[distant flume]

But how was I supposed to know that

[willow bear]

know what

[distant flume]

These formulas

Never seen them before

[willow bear]

i mean

you know how to find the volume of a box, right

[distant flume]

base times height

[willow bear]

length * width * height

or base area * height

in this case the height is h, and the base area is x^2

make sense?

[distant flume]

It does

[willow bear]

as for the area... you've got five faces

the base, which is a square of side x

and four lateral faces, which are rectangles of width x & height h

[echo plaza]

Not sure how this would be done without calculus

[distant flume]

I have done differantiationa and concavity but not much more, so I think this works

[limber bone]

what is the domain

of f(x) = -(x-6)^2+8

thats a polynomial

no?

[vestal plaza]

I guess idk the names

[limber bone]

domain of f^-1 here is R

where f^-1 is -(x-6)^2+8

[spring thunder]

is it?

[serene heath]

no, the range is y>=6

[spring thunder]

^

[serene heath]

so domain >=6

[limber bone]

ye i knew that but why tho

is it to make the inverse function work?

[spring thunder]

yas

[limber bone]

oh cool

sry

@vestal plaza

[vestal plaza]

do I just type 6?

cause it doesnt give me any option to type the >= part

[echo plaza]

tf

[spring thunder]

ft

[limber bone]

x greater than or equal 6

hahAHAH

[valid vector]

@wet hare Ask here

[wet hare]

Nice! 🙌

[full garden]

hey guys I have a small question

when I am proving a trigonometric identity

will I get the same answer if I work and the Left Side and not the Right Side, or theres a concept behind it?

[half axle]

https://gyazo.com/136917568b8f10e8f629dff129c60827

why is it -2ln(x-2)?

[thick raptor]

tfw post in multiple servers but never responded to help in other places

[serene heath]

lol

[barren hedge]

@half axle They want you to simplify as best as possible.

[wet hare]

I got doubts with how to get the period of this problem.

How did it get to 2pie/9?

[mighty pelican]

f(x) = Asin(Bx + C) + D formula for period is 2pi/B @wet hare

[wet hare]

Hmmm, okay thx

[viscid thistle]

Anyone on

Hey billie

kinda, was scrolling

Let me ask you a question

Have you heard of this

Book

Idk if I can help but sure

https://www.amazon.com/Pre-calculus-Demystified-Second-Rhonda-Huettenmueller/dp/0071778497

I'm taking pre-calc CLEP

Heard anything good about this book

never seen it in my life lol

CLEP is for?

to get credits for college or?

Yeah college credit

Are you good with pre calc

: D

I took it a long time ago

got to calc 3

based on the reviews it seems like its a good book

and its cheap

Nice

That's outstanding

pain in the a** though lol, too many proofs

Wait

So your computer scientist?

no engineering

Oh okay

Do you have to be

had to take some CS class though

Amazing at math for computer science?

you prob need physics 1 and 2 for engineering, calc1-3

I'm getting closer to college (im 11th grade) so im thinking about options and career choices

oh i need

calc 1-3

to get a CS Diploma

for computer science

yea

oh damn

im fucked man

nah its not too bad

Ay man if I push through calc 1-3

have you started to code yet?

No

Kind of

Kotlin

But it seems rather difficult