precalculus

[thin merlin]

Hello, does anybody know how to solve this question in exact form?

I'm supposed to code a calculator that would solve this in exact form, but I don't remember the math used to solve for exact form.

[viscid thistle]

Why in c) is the answer x+5 in the denominator instead of ×-5?

[sage urchin]

https://gyazo.com/2a83fe756cdf6f28e3b973a5243a2b7a

[tepid cloak]

help

[mild swan]

Drawing a picture will likely help

If you know vectors, you can use the dot product or cross product maybe

If not, you can use the fact that altitudes are perpendicular, so the product of the slopes of the line it touches and the altitude itself is -1

[hushed sphinx]

When you move the graph 5 units to the left, the vertical asymptote ends up at x=-5. So that's the x for which the denominator should vanish. That's true for x+5, but not for x-5.

[viscid thistle]

precalculus topics

[vast jewel]

-6e+3f=1 , 6e-4f=3 can someone help me with this equation using elimination method

[gleaming owl]

Add both.
6e cancels out, f=-4
Substitute value of f, get e

[viscid thistle]

How do i get cos^2x to become cosx

[hushed sphinx]

Divide it by cos(x).

[viscid thistle]

Yh but tanx =sinx/cosx so i need the cosx to be at the bottom dont I?

[hushed sphinx]

Remember $\frac{a+b}{c} = \frac{a}{c} + \frac{b}{c}$.

[obsidian monolith]

Troposphere

[viscid thistle]

Ohhk

Thank you

[autumn marsh]

yo my friend is stuck on a diff version of a question i did

so shes tryna find the ref angle for 7 radians

7 radians is in quadrant 1

i thought maybe it would just be 7 radians or 1 radian but neither worked

mine was find the ref angle for 4 radians and i figured it out but no idea

[mental narwhal]

how to solve this? Please and Thank you

[willow bear]

@autumn marsh it's 7 - 2pi

[void lintel]

can someone please prove Wilson's theorem?

with only high-school level math?

[cinder iron]

hi can anyone help with 4b and 4c? i got the correct answer to 4a but i think my diagram might be missing something?

[vapid plaza]

I’d suggest you check out numberphile’s video: https://youtu.be/AiplrfFB6h0

[viscid thistle]

Can someone explain what this is please?

[willow bear]

looks like the number-line method for solving rational inequalities

with a little bit of plug-and-chug to determine the sign on each interval...

[visual wave]

stuck on this part, could use some help

[hushed sphinx]

How did you get the 7?

[visual wave]

i actually dont remember was doing this 30 mins ago

i left the question now im back at it

[hushed sphinx]

Then your first troubleshooting step should be to calculate it again and see if you still get the same result.

If you do get the same result, show the work and we can figure out what's wrong with it.

[visual wave]

16000 makes sense because that is the total amount of given money

Q 14 is the work I got

I guess, I got 220 some how by dividing 440 by 2, genuinely I'm lost

[hushed sphinx]

(Repeating the main advice).

[quartz grotto]

Since when is the v written like it is on the x-axis? Lmfao

It needs to be (v, t), but it is a minor detail lol

[hushed sphinx]

Huh? The horizontal axis is t; the vertical axis is v.

[visual wave]

i got it, it was 20, there was litterly 3 mins left before submission

i started taking guesses

[hushed sphinx]

Hmm, curious; the actual break-even point should be at 19.00249 which I'd expect to round to 19 instead of 20.

[visual wave]

no idea

that is too much of a wiggle room

[hushed sphinx]

FWIW, it should just have been a question of solving the quadratic equation -2x²+880x-16000=0.

[visual wave]

ahh

[opal garnet]

Hello

Can I get some help here?

[stuck lark]

[opal garnet]

ok

i had this question

i knew it is some number theory question

but then i saw we can make a function with two variables

and then if we take grad of that function and work out

we get the answer as 160 itself

but i am sure it isn't the answer because the answer is of 2 digits

so i need to know where am i going wrong

[willow bear]

show work?

it might be that you are computing the largest value of x + yz instead of the smallest

[opal garnet]

ok sure

just give a minute to find it

there you go

[willow bear]

,rccw

[willow bear]

this is supposed to be an x?

[opal garnet]

yea

[willow bear]

without sugarcoating your handwriting sucks lol

[opal garnet]

ik that lol, but i really don't care until they are my notes

do you want it to be more 'fair'?

[willow bear]

i'm trying to figure out exactly where you're going wrong here

what's the domain of your function here
x ≥ 1, y ≥ 1, xy ≤ 159...?

[opal garnet]

i think there are no restrictions other than being positive

[willow bear]

don't forget that z has to be positive too

[opal garnet]

yea at the point where x and y are what i've written, z is

[willow bear]

this feels weird to me

wait

[opal garnet]

idk i just let my mind let run wild

and stumbled upon this

[willow bear]

have you checked the hessian at the point (1, 80)

sure it's a stationary point

[opal garnet]

well, what's that?

[willow bear]

but it might be a minimum or a maximum or a saddle point

the hessian is the matrix of 2nd derivatives

[opal garnet]

and what does it do?

[willow bear]

have you heard of the 2nd derivative test for max/min points

it's the multivariable version of that

you look at the hessian matrix at your point and calculate its eigenvalues

if they're both negative you have a max point, if they're both positive you have a min point, if they're different signs you have a saddle

if one of them is zero you are fucked

[opal garnet]

i remember what it is at the back of my mind

[silk escarp]

Guys can I be doing this?

What is confusing me is can I divide both sides by -2?

[gleaming owl]

You gotta find value of x right?

[vivid void]

why wouldn’t you be able to do that?

[silk escarp]

The thing is I'm trying to go back from the complex solution and it leads me to two different places. Both are started from x = 3 - 2i

[willow bear]

you got two equations which differ from each other by a multiplication by 4 and nothing else

[silk escarp]

Oh my God

Ann you rock!!

[vivid void]

bruh

[solar cradle]

multiply everything by cos^2 and that should get you somewhere

[supple valley]

I forgot how to do these types of problems, any help?

[hushed sphinx]

Draw a right triangle with theta in one corner. The given value gives you two of the side lengths; pythagoras will supplly the third if it turns out you need it.

The result you want is now the ratio of two appropriately selected sides.

(Note that pi/2-theta is the measure of the other non-right angle in the triangle.

[supple valley]

Oh right thank you

[mossy ermine]

is the +1 with cos^2 or is it tan^2 + 1

[placid knot]

@bold zinc please don't spam, you are muted for 24 hours

[silk escarp]

Hey guys, say you have a polynomial, it has linear factors(x intercepts) and irreducible quadratics ( complex zeros),

So I read this thing about the relation between complex solutions and turning points in the polynomial graph.

Could someone please tell me if this statements is true

"number of turning points = number of complex zeros/solutions"

[trim hemlock]

not really

excuse me, that's absolutely false

a polynomial of degree n will have n complex roots

however can have at maximum n-1 numbers of turning points

[silk escarp]

Got you

[viscid thistle]

i have 4 questions that i cannot do, can someone help, they all have to do with polynomials

[full pagoda]

[viscid thistle]

guys umm I have a question, can u help me? write the trig form of -7+4i
thank you

[vivid void]

you mean polar form i guess

[viscid thistle]

hi can some one help me with exponential functions

[hushed sphinx]

[steel fern]

In precalculus in my school there are 2 classes. l in algebra and one in geometry

I am looking for more advanced geomerty books

to teac hmy self

[tight compass]

I was planning to go through this https://www.math.ttu.edu/~rgelca/GEOMETRY.pdf to review my high school geometry in some point in the future.

[viscid thistle]

You mean $e^x$

[obsidian monolith]

CyberAnthrax

[viscid thistle]

The idea:something always grows in relation to its current value, such as always doubling

Example: If a population of rabbits doubles every month, we would have 2 then 4 then 8,16,32,64,128,256 etc

Let us say that we have a special tree

It grows exponentiallyfollowing this formula

Height in mm = $e^x$

[obsidian monolith]

CyberAnthrax

[viscid thistle]

e is Eulers number, about 2.718

At about 1 year old the tree is $e^1$

[obsidian monolith]

CyberAnthrax

[viscid thistle]

which is 2.7 mm high…really tiny

At 5 years old it is $e^5$ =148mm high as high as a cup

[obsidian monolith]

CyberAnthrax

[viscid thistle]

At 10 years old the tree is $e^(10)$
= 22m high as tall as a building

[obsidian monolith]

CyberAnthrax

[viscid thistle]

of course such a tree couldnt exist

so in other words just think of it as growing exponentially

i hope this helps you

[viscid thistle]

To learn more please visit www.mathisfun.com

[lean hill]

Let $P(x)=x^2+t$ for some positive integer $t\ge2.$ Define sequence $x_n$ as $x_0=0$ and $x_n=P(x_{n-1}).$ Then prove that there are infinitely many $n$ such that $n$ divides $x_n.$

[obsidian monolith]

AeroBennu

[willow bear]

this looks more like number theory than precalc ngl

@lean hill

[viscid thistle]

Looks more like a lemma to me

[viscid thistle]

Gotta write a test now seeya

[timid cedar]

I think this question fits here...

$\frac{e^3x}{e^2x} = y$

[obsidian monolith]

Mallot

[timid cedar]

how do you solve for y? How goes the e rules here? do I just take ln on each term, or what do I do? I can move over the e2x to the right, and then take ln on both sides... do I then get 2x ln (y)?

[bold meadow]

y=e^x

just use rules of indices

[willow bear]

how do you solve for y?
this equation is already solved for y.

also, in case you meant for the x's to go in the exponent: $e^{2x}$

[obsidian monolith]

Ann

[willow bear]

@timid cedar

[timid cedar]

Yeah, switch places with y and x

[willow bear]

so what you actually wanted to write is this?

$\frac{e^{3y}}{e^{2y}} = x$

[obsidian monolith]

Ann

[timid cedar]

yeah

[willow bear]

so $e^y = x$

[obsidian monolith]

Ann

[glass relic]

Hey, help me with the 19 please

[stuck lark]

we discourage exchanging money for services here

also just post ur question

[crystal aurora]

The first inequality is a shaded circle
Second one is like an upside down symmetrical triangle with a point at the origin.

Since the the gradient of y= |x| is just 1, we can deduce that the angle between the line and origin is pi/4

Hopefully thats good enough to get you going

[glass relic]

Thanks!

[delicate edge]

Does anyone know how to solve ths

[rose basin]

Anyone know how to solve this composition function right here? The answer is what confused me a little bit when I worked through this homework problem. How is it -2 at the end of the answer is my question? I understand why it's t^2+2t but I'm confused about the -2 at the end.

[spice shale]

I would think it would be finding a domain that keeps x^2-x > 0 because you can’t have a log of a negative or of 0 I don’t think

[uncut mulch]

show what you did @rose basin

[uncut mulch]

your error is that you're missing the 1 in -4(1-t)

[rose basin]

I see now thanks

[delicate edge]

okk

[kindred bluff]

general question but for euler's theorem and Fermat's little theorem, do both numbers need to be natural numbers?

[hushed sphinx]

Yes.

(Well, the base can be a negative integer if you want, but it's taken modulo something, so that's not really an extension anyway).

[viscid thistle]

i despise pre calc

[jolly raven]

its a very despisable class

how come you despise it tho

[viscid thistle]

i dont like the way my teacher teaches

she gives us video notes from i'd say 8 years ago based on the quality and off of that we take notes at home. in class, we do our "home"work or take tests which is basically every week. The way she teaches is very confusing to me and hard to understand

so last couple of months, i just stopped doing the notes and started watching videos on youtube instead, which definitely raised my grades and helped me understand wayyyyy better

[jolly raven]

if you enjoy math, i promise calculus will feel like a huge landmark in your knowledge

i just finished calc bc today and its really nice to take a step back and realize ive learned all of the high school math stuff

which is really weird

but yeah

math is really rewarding

[viscid thistle]

i hope so, ive been fine or liked math until this class

[jolly raven]

dont worry

if you like learning about new and really conceptual interesting things

then calculus is for you

[viscid thistle]

neat, looking forward to it

now im kind of regretting choosing normal calculus

[jolly raven]

hmm

do you want me to show you the idea for the driving concept of calc 1?

well

have you learned what a limit is yet

[viscid thistle]

yeah

if its the same limit that im thinking about then yeah

does it have to do with graphs

[jolly raven]

yeah kind of

so have you ever seen this

$\frac{y_2-y_1}{x_2-x_1}$?

[obsidian monolith]

hiidostuff

[jolly raven]

or

$\frac{\Delta y}{\Delta x}$

[obsidian monolith]

hiidostuff

[viscid thistle]

delta symbol

no i have not

[jolly raven]

the top one?

[viscid thistle]

i dont believe so

[jolly raven]

you havent ever seen any of these?

[viscid thistle]

well I've seen it

[jolly raven]

have u used it in school?

to find slope

[viscid thistle]

I have not, we probably used some different method

when do you learn this usually

[jolly raven]

algebra 1

when you do $y = mx + b$

[obsidian monolith]

hiidostuff

[viscid thistle]

ohh, yeah i've used it

[jolly raven]

the m that you see is difference in y divided by difference in x

[viscid thistle]

in like 7th grade

[jolly raven]

yeah see

u know

so what a derivative does

have you ever heard of a derivative?

like just heard it when someone was talking

[viscid thistle]

yeah i used it to find an equation for a line thats tangent to a parabola

only once that time though, i watched a youtube video for it

[jolly raven]

ayy

thats what im about to show you

do you remember what the slope of the tangent line to the parabola was?

[viscid thistle]

yeah

[jolly raven]

what was it

[viscid thistle]

it was something like y = 4x + 14

[jolly raven]

ok

so you were probably working with the parabola

2x^2 + 14x + c

something like that

[viscid thistle]

actually imma go get it real quick

[jolly raven]

ok

lemme work on a graph

[viscid thistle]

ok so

this is how my teacher solved it

[jolly raven]

nice

[jolly raven]

so lemme expand that equation

[viscid thistle]

this is how i solved it

its a little cursed, as its my first time using derivatives

[jolly raven]

so it works out to be x^2 + 12x + 36 = 3y - 6

you did it well tho

did you learn the definition of a derivative?

or just some rules

[viscid thistle]

i have no clue what it even is, i just know that i used it haha

[jolly raven]

alright

well im about to show you

[viscid thistle]

yesss

[jolly raven]

so in the past what have u done to denote slope for lines?

[viscid thistle]

what do you mean denote slopes?

[jolly raven]

so what did you define the slope of a function as

or just a linear function

[full dock]

he means like rise over run

[jolly raven]

ye

something like this

[viscid thistle]

im not sure what its called hold on

let me look at stuff from last year

[full dock]

simply put, derivatives are a quick trick to finding the slope of a tangent line to a curve

[jolly raven]

yeah

[full dock]

im sure hiido will explain more

[jolly raven]

they are a really cool trick

and very useful

[viscid thistle]

oh so i used it correctly

it was literally meant for that problem

[jolly raven]

yeah

[viscid thistle]

is it always simple as the one i sent?

[jolly raven]

hmm

gimme a sec

[full dock]

derivatives are usually easy enough, unlike integration

[jolly raven]

well

[viscid thistle]

i heard that integral is pretty hard

[jolly raven]

derivatives typically are tedious rather than difficult

[jolly raven]

theres a lot of them that we simply dont have enough math to solve

thats a common thing that appears in diffeqs

i dont wanna clutter precalc chat tho so ima move to dms andy

[viscid thistle]

sounds good

[full dock]

derivatives are "easy" since we have easy formulas like the product rule, chain rule and quotient rule, integration doesn't have those explicit formulas

[jolly raven]

^

[viscid thistle]

oh so they're usually easy but annoying to do

[full dock]

yeah basically, ill let hiido take it from here

[viscid thistle]

yeah he gave me a good explanation

[sick steppe]

Unknown

[willow bear]

for f(x) = 19(x^420 + 69) there is no such B

@merry plover

[tame grotto]

Shush

[undone sandal]

does any1 know how to do these??

[nocturne iris]

for the first one

subtract 3sec^2(theta) on both sides

then subtract 1

then you have a quadratic

solve for it

then solve for theta on that interval

do the same for the others

[undone sandal]

ty

[visual wave]

im doing one of the questions for my h.w problem for functions, and its not adding up in my head

to me that looks correct, the factor of 7 should be outside of the Square root and the -7 should not be inside the square root because its being shifted down

[willow bear]

you scaled vertically instead of horizontally

[visual wave]

oh is that..

[willow bear]

sqrt(x/7) - 7 is what you're looking for.

[visual wave]

so im just dividing x/7 meaning bringing the 7 thats outside the square root inside got it

ill make a note of that, Thank you

[frigid haven]

To scale horizontally you multiply the x itself by a number. Numbers bigger than one scale down and smaller than one scale up

So in this case you would go from $\sqrt{x}$ to $\sqrt{\frac{1}{7}x}$

[obsidian monolith]

lexitorius

[frigid haven]

Or x/7

[unkempt bay]

can someone help out?

[willow bear]

@unkempt bay do you still need help with this?

[unkempt bay]

noper

i got it

😄

[torn zinc]

https://www.desmos.com/calculator/e2nnjcaytc does this graph match these requirements?

[unique oxide]

can someone help with A?

I need to verify it

[topaz roost]

Not sure if you have already been helped or not but I decided to try and work this one out and this is how I did it

Putting everything in terms of sin and cos is generally really helpful when trying to verify identities

(Also sorry I have terrible hand writing 😅)

[calm burrow]

Does anyone have practice problems for either asymptotes, complex zeroes, or real zeroes of a polynomial function? I've got a PreCalc test on Thursday that I need to prepare for

[finite heath]

Polynomial functions or rational functions?

Because polynomials aren't really gonna have asymptotes

For zeros, try x^4-4x^2+3x+2

I don't have like a whole set but that's a pretty interesting problem

[spice shale]

an asymptope would really only come if you had a polynomial (or other function with a zero) on the denominator

[finite heath]

Aka rational yes

[spice shale]

oh lol i didnt see your message

[vagrant ingot]

can someone help me pls

given tanθ=5/x
find cosθ

[jolly raven]

what can you express tan theta as

[vagrant ingot]

sinθ/cosθ

[jolly raven]

yep

so now u just need to isolate cos(theta)

[vagrant ingot]

but how would i incorporate tanθ=5/x

[jolly raven]

well i mean

all u did is express tan theta as something that includes cos theta

so just isolate it

isolate cos theta

[vagrant ingot]

oh wait im stupid

i was trying to over complicate this

[jolly raven]

yo i overcomplicate stuff all the time

dont worry about it

[vagrant ingot]

i dont think thats how u figure it out @jolly raven

[jolly raven]

i dont see how else u would

[vagrant ingot]

its something with the unit circle

[jolly raven]

ohhhh

you have to know what the coordinate ratio for tan is

like how sin is y/r

[vapid plaza]

I think you can apply
tan^2(x)+1≡sec^2(x)

[jolly raven]

not at all whats going on here

[vapid plaza]

Why not?

[jolly raven]

so i just realized

what he means is that the trig functions can be represented with ratios of x, y, and r

like on the unit circle

[vapid plaza]

Well ||constructing a RH triangle with arms 5 and x|| trivialises the problem anyways

[jolly raven]

exactly

[vapid plaza]

But it may miss some negative solutions for cos

Because geometry

[jolly raven]

eh

i think positive is what we looking for

[vapid plaza]

Hmm
answer should be just ||x/sqrt(25+x^2)|| then

[vagrant ingot]

this is confusing

[jolly raven]

did you learn stuff like $\sin{\theta} = \frac{y}{r}$?

[vagrant ingot]

i never done one with an x before

[obsidian monolith]

hiidostuff

[vagrant ingot]

yeah

[jolly raven]

what is cos then

[vagrant ingot]

we used dif letters

a^2+b^2=c^2

u talking ab that

[jolly raven]

ehh yeah

but x y and r are more rigorous imo

so x^2 + y^2 = r^2

[vagrant ingot]

ok

[weary aspen]

Hi guys! If anyone could help me with some of these simple pre calc problems I’d be very thankful!

[vagrant ingot]

cos = x/r

[jolly raven]

https://www.youtube.com/watch?v=RuXPe8onmfM

[jolly raven]

yep

so whats tan

[vagrant ingot]

y/x

[jolly raven]

nice

so now with tan theta = 5/x

ur given a y right?

[vagrant ingot]

no thats the thing

if i was i could do it

[jolly raven]

you literally are given a y

[vagrant ingot]

im not

[jolly raven]

5/x = y/x

[vagrant ingot]

ill take screen shot rn

[jolly raven]

whats y

[vagrant ingot]

wait

[jolly raven]

yeah see

u gotta substitute

[vagrant ingot]

bruh

[jolly raven]

so whats ur y

[vagrant ingot]

let me take a screen shot rq

[jolly raven]

i see the problem dude

but i promise you you are given a y

you just have to be willing to do the algebra

so tangent can be rewritten as y/x correct?

[vagrant ingot]

wait so y=5

[jolly raven]

yeah

see

so

now with cos theta what 2 variables do you need to find

[vagrant ingot]

x/r

so opp/hyp

[jolly raven]

ye

but lets stick with x/r

so i think here you wanna represent r in terms of x

so what function can you use

that has everything to do with triangles

that involves x y and r

[vagrant ingot]

this

[jolly raven]

ye ye

so now substitute our value in for y

[vagrant ingot]

but idk how i can do that with too unkowns

[jolly raven]

well if you express r in terms of x

theres only one unknown

and we can just leave it at that

[vagrant ingot]

would it be x/(r+5)^2

[jolly raven]

hold on

[vagrant ingot]

or would it be (x+5)^2/r

[jolly raven]

neither

you wanna have r = f(x)

f(x) just being any thing represented by x

[vagrant ingot]

ye

[jolly raven]

so with x^2 + y^2 = r^2

what value do we know

[vagrant ingot]

y

[jolly raven]

yep

so now its x^2 + 25 = r^2 right?

[vagrant ingot]

yeah

[jolly raven]

how can we isolate r here

[vagrant ingot]

like get rid of ^2

sqrt the other side

[jolly raven]

yep

so now we have $\sqrt{x^2 + 25} = r$ right?

[obsidian monolith]

hiidostuff

[vagrant ingot]

yeah

[jolly raven]

so we have something in terms of r

cos is x/r

so now just replace the r part

with the new x stuff

[vagrant ingot]

ok

[jolly raven]

so what would that be

well

its gonna be hard to write

ill do it for u

[vagrant ingot]

wait do i put two xs

[jolly raven]

$\frac{x}{\sqrt{x^2 + 25}}$

[obsidian monolith]

hiidostuff

[jolly raven]

did you get this?

[vagrant ingot]

yes

[jolly raven]

nice

that should be it

[vagrant ingot]

ill let you know when he grades it

[jolly raven]

alright bro

[vagrant ingot]

thanks a lot

[weary aspen]

Can anyone help me with this problem? It’s timed and I keep running out of time doing the other problems, but I need all parts correct for credit

[vagrant ingot]

a is slope intercept form

and then u just plug the stuff in

id recommend desmos

[slate quarry]

Is it possible for me to take the natural logarithm of summation of e's?

[willow bear]

what do you mean

[slate quarry]

sorry if it's not totally precal related, I just figured since I'm talking about summations then it could be part of the topic

P is unknown tho I just wanna know if there's any way I can isolate P in one side by manipulating the summation

[cedar bramble]

can someone help me with this

sec(10x-10)sin(2x^2)=1

[finite heath]

sin(2x^2)=sin(10x-10)
2x^2=10x-10+2pik AND 2x^2=pi-(10x-10)+2pik

It's basically just a bit of algebra from there

("and" here really should be "or" because it'll hold if either side is true, I just said and to emphasize that you need to solve both)

[rapid sorrel]

can some one help me with the top pic

[vale helm]

Guys, I dunno if this question is appropriate but uuuuh

I know that 9 is a power of 3

but can we consider 3 to be a power of 3?

[finite heath]

Yes

3^1=3

[jolly raven]

Yep

If 3^n = x where n is an integer

Then x is a power of 3

[viscid thistle]

someone help please, with all of the question

for the first one i got -18

i think the last is DNE

but i have no clue for the middle one

-1 for the middle??

[full pagoda]

yeah

[viscid thistle]

is it right?

[full pagoda]

yeah

[viscid thistle]

okay thank youu

can u also help me with another question pls

i thought it was -infinity

but it wrong

[vivid void]

conjugate

[viscid thistle]

nvm i got it now

its 1/5

[rose basin]

So I'm wondering is my answer to my even number homework question correct, P(Q)=4-1Q. Or is it wrong?

[vivid void]

it’s wrong

[rose basin]

So how do you solve it

[vivid void]

assume it’s in the form of y = mx+b, then find m and b

[rose basin]

Alright

[viscid thistle]

Hi, need help checking this question
Find the exact value of each of the remaining trigonometric functions of theta
cot theta = - (sqrt3)/7, given that theta is in quadrant IV

I'm looking for sin, cos, and tan. So far I have that
sin = 0
cos = 1
tan = 0

appreciate any help

[sick steppe]

why assume? It is in that form since 1st differences are constant

[sick steppe]

how is cos(t)=1 if you know cos(t)=-sqrt(3)/7?

[vivid void]

i know it’s in that form, idk why i said that

[viscid thistle]

the given is cotangent, not cosine

[sick steppe]

oh misread

but how is cos(t)=1 then?

[viscid thistle]

I got that x and r are the same, so cos = x/r = 1

[sick steppe]

🤨

How is the hypotenuse of a right triangle the same length as a leg

[viscid thistle]

good question

[sick steppe]

Yeah, double check your calculation

you know that $x=\sqrt{3}$ and $y=-7$

[obsidian monolith]

Mosh

[viscid thistle]

I got that r = sqrt52

[sick steppe]

Yeah

2sqrt(13)

[viscid thistle]

so I just need to plug them in and simplify them now, right?

[sick steppe]

yes

[viscid thistle]

alright got it, thanks

[sick steppe]

since you know x,y and r, you know all the trig ratios

[viscid thistle]

just making sure i'm doing it right, the given is an exact value right? so for example tan theta would be

• (7sqrt3 / 3 )

[sullen lichen]

@mild swan So basically

You know what root -9 can be simplified into like

3I

3i

Since 9 is a perfect square

[mild swan]

Yup this is right

[sullen lichen]

Can we simplify root -10 for example with complex number

[mild swan]

A little bit

[sullen lichen]

Do we keep it as root -10

[mild swan]

You can’t simplify √10 though

[sullen lichen]

Or like root 10 i

Root 10i right?

[mild swan]

This

[sullen lichen]

ty @mild swan

[mild swan]

Only the 10 is in the √

[sullen lichen]

!rep @mild swan

[mellow tapir]

i * sqrt(10) remember

[sullen lichen]

Ok!

[mellow tapir]

its t!rep _____

[mild swan]

👀

[mellow tapir]

and you might hav to make an account

[sullen lichen]

t!rep @mild swan

What ummm

[mild swan]

?

[mellow tapir]

do it in #bots

[mild swan]

Lol Ty Ty

[sullen lichen]

Hi

@vivid void What exactly is a conjugate and what's the point

Oops wrong person

My bad.

[sullen lichen]

I am learning conugate in complex number, it's just switching signs

How come the conugate concept is in complex numbers and not in like normal algebra

[mild swan]

The conjugate of a complex number a + bi is a - bi

[sullen lichen]

What's so special about conjugate in complex number

Like we never learnt a + b = a-b

[mild swan]

The product of a complex number and its conjugate is a real number

[sullen lichen]

What exactly is a conjugate

[mild swan]

Im not exactly sure as to why the conjugate was made, but I know that this is one of its properties

[sullen lichen]

What's the point of know that btw, is it useful

[mild swan]

I think it’s just defined as (what I said above)

[sullen lichen]

Okay

[mild swan]

You can probably Google more about it

[sullen lichen]

Okay!

[mild swan]

It can help with simplifying complex numbers

[sullen lichen]

Okay!

Btw I am learning to multiply and divide complex numbers now

I am revising everything for the test

[sullen lichen]

Are you from india because your good at maths

[mild swan]

Lol nah

[sullen lichen]

Australia

That's where I am from

[mild swan]

Ik you said that already

[sullen lichen]

Did I?

[sullen lichen]

Is it basicallly like rationalizing

[mild swan]

#calculus message

[mild swan]

That’s actually a cool way of thinking about it 🙂

Yup

[sullen lichen]

Is there a point of doing it

If the complex number is in the numerator

And the real number is in the denominator

[mild swan]

Of rationalizing?

[sullen lichen]

Is there point of multiying the numerator which is the complex number by the conugate

For simplifying

By rationalizing

Yeah

If it likes a+ bi / 3

Do we multiply it by a-bi / a-bi

Since the complex number is in the numerator

[mild swan]

You’d normally use multiplying by the conjugate if you have a complex number in the denominator

[sullen lichen]

Or is it just like rationalizing

Okay

If it's in the numerator, nothing happens like normal rationalizing?

[mild swan]

Same with rationalizing, you’d only do it if you have an irrational number in the denominator

Like a √

[sullen lichen]

Thank you!

[mild swan]

The purpose is to clean up the denominator, not the numerator necessarily. Therefore, you may not get a SUPER “clean” number for the numerator.

Sometimes you will, sometimes you won’t.

[sullen lichen]

Okay!

[sullen lichen]

I am confused about this thing but we can talk about it later when I revise it

My teacher said we should know like exact form off by heart

To do like polar form and stuff

Does she mean like the tan(0) = pi/x

Or does she mean like tan(0) = root 2/ root 2

This is just an example, does she mean know it with the pi value of the theta

[mild swan]

Those aren’t necessarily true, but maybe? I’m not sure

Exact form of what

[sullen lichen]

I know all the sin/cos/tan off by heart

Like these are the questions I am gonna do later

[mild swan]

Ah okay

[sullen lichen]

I haven't understood this area of the topic much yet, I know like sin/cos/tan 0 30 45 60 90 off by heart, will it help me with this or it is a whole different story?

[mild swan]

If you know what cis(θ) is, you should be good

[sullen lichen]

I will learn it once I finish other excercises/revise

What exatly is cis?

[mild swan]

Because you’ve memorized them

[sullen lichen]

Yes but I don't know like

cos(pi/4)

I am not good with pi

Is there a way where I can transfer the cos sin tan into pi

[mild swan]

I thought you said you knew all of the sin cos tan by heart

If you only know it in degrees, you can convert radians to degrees by multiplying radians by 180/pi

[sullen lichen]

@mild swan Are you here

Still

[sullen lichen]

Can you use distrubutive law in complex numbers

like

(4+i)^2

i mean can u use the perfect square

[willow bear]

yes, complex numbers still obey the distributive law

and yes, you can use the identity (a+b)^2 = a^2 + 2ab + b^2 even when a and b are complex

[sullen lichen]

@willow bear Ttsn

\

]

[tight compass]

cosx + isinx.

[sullen lichen]

@tight compass do we need to write that

is it necessayr to write cis

[tight compass]

it just a notation for that.

you have to compute it with that.

does angles should be familiar with you, so it should be easy enough.

[sullen lichen]

tysm bro @tight compass

[silk escarp]

If a log has a negative base or negative argument alone it doesn't work out, if the both are negative it seems to make sense

Could this be done?

[hushed sphinx]

You can find solutions for certain particular values, but they do not combine to form a continuous logarithm function, so we don't usually use that notation..

[frigid wagon]

The image of a function is basically like a restricted range with a restricted domain right?

[hushed sphinx]

You can view it that way

That would be the image of a subset of the domain.

The image of the function itself is just a synonym for its range.

[frigid wagon]

yeah I mean a subset of the domain

and a pre image is just an image but reverse right

[hushed sphinx]

Basically.

[frigid wagon]

unless some elements of the set being pre imaged are not part of the range

yeah

ok thanks

[sullen lichen]

<@&286206848099549185>

[hushed sphinx]

Please don't ping Helpers immediately.

[sullen lichen]

OK

[hushed sphinx]

Can you divide complex numbers?

[sullen lichen]

Yes we can

[hushed sphinx]

Then just divide both sides by 2+5i.

[sullen lichen]

k

smart

@hushed sphinx can u try doing it by expanding brackets

it wouldnt work for me

'could u try

[hushed sphinx]

Why do that if you can just divide?

[static thistle]

Take the derivative of top and bottom

Then evaluate

[sullen lichen]

@hushed sphinx i might not realise it in exam

could u try it

to see if it works

[hushed sphinx]

Sure, expanding the multiplication to end up with two linear equations in a and b will work too.

[sullen lichen]

@hushed sphinx couldl u do it

i couldnt do it

[hushed sphinx]

I am comfortably confident that I can, but I'm not going to do your homework for you.

[sullen lichen]

@hushed sphinx u cant

try it

and its not homework

its my own study

<@&286206848099549185> need somene to help me

[balmy swan]

Just do what Tropo suggested, or review multiplication of two complex numbers

[sullen lichen]

@balmy swan i did bro

[sullen lichen]

What does c mean?

[obsidian monolith]

Alter

[mild swan]

@sullen lichen ^^

In other words, solve for all complex solutions.

[sullen lichen]

@mild swan tysm

can u help me do this

im stuck on f

[mild swan]

Quadratic formula

[sullen lichen]

i tried it

could u help me once with this

it wont work

i cant get the answer in the book

[hushed sphinx]

What do you get, and how?

[tropic fossil]

this isnt precalculus but its review is this correct?

anyone here?

[hushed sphinx]

As you said yourself, you're in the wrong channel. Try #calculus.

[tropic fossil]

can u go there and answer my question?

[sullen lichen]

@mild swan hi

[sullen lichen]

Can someone help me with 1d

I got

4root2 CIS -1pi/4 as my answer

[uncut mulch]

show work

[mild swan]

@sullen lichen ^^

[sullen lichen]

@mild swan oh im getting help lareafy in different serevrer

[mild swan]

Man’s drunk hehe

[sullen lichen]

@mild swan anyway extra help will do

[mild swan]

For?

[sullen lichen]

@mild swan question 1c

test tommorow 😭

[mild swan]

What have you tried so far?

[sullen lichen]

so basically

i dont really get this sin cos thing

i got the r is 4

so

4(cos -1root3/4) + isin 1/2)

i don't know what to do next

[mild swan]

I don’t understand your notation

Your complex number is r cis θ

[sullen lichen]

exatly

[mild swan]

This should be equal to your a + bi

[sullen lichen]

i cant find the theta

i found the 3

exactly

i found the r

which equals to 4

i cant find the theta

[mild swan]

Expand this and set the real and imaginary parts equal

[sullen lichen]

wdym?

[uncut mulch]

consider plotting the complex and/or
consider arctan with properties of the unit circle and/or
the arctan2 function

[mild swan]

Find the real and imaginary parts of r cis θ and set them equal to the real and imaginary parts of the complex numbers you’re given

[sullen lichen]

@mild swan i dont get it still

could u just me just for that question

i already polotted

its in the second quadrant

[mild swan]

Expand r cis θ

[sullen lichen]

how

rcis

[mild swan]

You know what cis θ is

[sullen lichen]

yes

[mild swan]

Now multiply it by r

[sullen lichen]

r (costheta + isintheta)

[mild swan]

You’ll get a real and a imaginary part

[sullen lichen]

ohhh now i get it

but how do i find

[mild swan]

Then this

[sullen lichen]

costheta

and sin theta

[mild swan]

This

[sullen lichen]

how do i find it though

[mild swan]

You set them equal and solve for θ .

[sullen lichen]

https://www.youtube.com/watch?v=Xb8f3CHchiI

this youtube helped me ty

[glass crag]

Would someone be willing to share their brilliant account with me? I don’t exactly have the money to pay for a calculus course but I would really like to do it

ping meh if u wanna

or shoot me a dm idm

[timber obsidian]

z = re^i theta
baso, for anything in the form z= x+yi
r = sqrt(x^2+ y^2)
tan(theta) = y/x

remember tan theta has 2 values that satisfy the for every 2pi, pick the right theta using common sense of which quadrant it lies

so if both x and y are positive, theta : 0<theta<pi/2
if they’re both negative
theta: + or - pi of what you got initially for theta

@sullen lichen

[rose basin]

I'm wonder how to solve this problem. It tells me to find the parallel and perpendicular slopes with just m=2 and no other information.

[uncut mulch]

the question is worded poorly

but they seem to mean to:
a line with m=2 (i.e a slope of 2)

[woeful acorn]

Hey guys, quick question about matrices, particularly transformation matrices. I read the articles on khanacademy and watched a video from 3b1b. They explained what the basis vectors are and how a vector can be defined using a combination of only these two base vectors. They later explained how after applying a transformation matrix, the original vector which was a combination of the basis vectors is now equal to a combination of the TRANSFORMED basis vectors. My question is how are the basis vectors transformed in the first place? Say we have the transformation matrix:
| 2 , 1 |
| 1 , 2 |

How do the basis vectors get transformed from 0 , 1 and 1 , 0 into 2,1 and 1,2 respectively? Multiplication can't yield that result since anything multiplied by zero is just zero. And addition would shift the origin therefore turning the transformation into a non-linear transformation, so how does that work? I think I'm misunderstanding the concept.

[old isle]

the rate of change parallel to something, assuming it is a y=mc+x is thesame

unless that slope is curved, m=2 should apply to every parallel slope

[rose basin]

Alright thanks

[viscid thistle]

Is there any graphing calculator for complex functions?

[finite heath]

It's a little more complicated than you might think

With complex, all functions are in 4d, not 3d or 2d

Which means that our 3d brains can't really see all of it

But, vector fields exist, and they're the most common way to visualize functions like that

And yes, there are a bunch of vector field visualizers

Geogebra has one iirc, but I generally use a custom one on desmos. Whatever works

...if by "complex" you really meant "complicated", then just use desmos

[novel helm]

theres things out there that plot the modulus and argument

[viscid thistle]

No I mean with i and all that

Complex functions

[novel helm]

but yeah cant ""plot"" a line like a usual function since itd have to be 4d

[viscid thistle]

Ohk

But ive seen graphs with Y axis as ImaginaryNumbers…thats why I asked

[finite heath]

Yeah, you can graph one dof at a time (e.g. modulus, real part, etc)

[viscid thistle]

Ok is there a way to see the graph of
e^(ix)?

[finite heath]

But you can't graph everything without losing at least a little information

[finite heath]

Yeah, that has one dof of input. As long as you're aware that you won't be seeing any nonreal inputs, you can plug the conversion into geogebra 3d and see the graph

In this case it's just cis

[viscid thistle]

Ok

[knotty wolf]

hi guys :)

I'm a algebra 2 student who is learning precal to get a bit ahead

quick question

I was working on a complex plain in khan academy, and I don't see where I would have to use it other than graphing complex numbers?

[mild swan]

You can probably Google Applications of the complex plane

[finite heath]

Most applications aren't obvious but they come up a lot

Well maybe they are obvious if you're super familiar with them

[knotty wolf]

I just wanted to get someone who is actually studying it to explain it

since I have a deep mistrust of google

[finite heath]

I'm in complex analysis rn. There's no short answer

The easiest "direct" application would be rotations

[knotty wolf]

alright, I just wanted to know IF I would use it

so thanks mate

[finite heath]

Any complex number can be thought of as a combination of a dilation and a rotation, both about the origin

[finite heath]

Oh in precalc? Complex numbers are a single unit if they're included at all

Afaik

[mild swan]

^^

[austere hound]

Anyone who need help in pre calc on Aleks?

[fringe furnace]

pussy

[old wasp]

Yo guys, I really love maths so I moved onto calculus before I had mastered everything in precalc and things before it, and I would say I am quite decent at calculus (only single variable, calculus 1 level) but now I want to go back and master everything to make sure there are no gaps. What do you recommend I should learn before I go back to calc?

[viscid thistle]

trigonometry probably

[glacial mirage]

And stuff with graphs, as a British student I don't really know about calc 1, 2, 3, etc. but I image graphs, such as graph transformations, like f(x) mapped to f(x+5) for example, inverse graphs, logarithms, like e^x, ln(x), log base n, etc. etc.

If you get stuck, TLMaths is a good website, mainly for A Level and Further Maths in the UK, but you should be able to find what you're looking for quite quickly

https://sites.google.com/view/tlmaths/home

[glacial mirage]

Actually, just reading this back, having a look at some practice questions for precalc stuff for British papers, not looked at American papers that much but British ones have a lot of applications rather than regurgitation of knowledge, PaMT should have some questions, and they'll have some exam questions as well which will definitely help with the understanding

https://www.physicsandmathstutor.com/maths-revision/a-level-aqa/

[lean socket]

Hello guys,I'm a university student.i am struggling financially ,I was asking if you guys could link me with a premium discord server @everyone

I'm super at math

[mild swan]

The best we can do is just answer your questions when we have time

[lean socket]

@mild swan 😩

[somber yew]

You can ask problems in an appropriate channel on this server when you're stuck, you don't have to pay for it. All that is expected of you is to show some effort in trying the problems and cooperating when someone's helping out. Read #❓how-to-get-help for info.

Also, please don't use the everyone ping.

[swift flume]

why is this not 6C2?

[viscid thistle]

6C2 means you are choosing 2 from a bunch of 6. You need to form a committee of 6, not 2

[swift flume]

u right

wait

no

but I already know there's only two men

so there's two men and 4 women

so is that not 6c2 ways of picking that

[outer briar]

how it became negative

[gleaming owl]

it's stated in question, that angle is in second quadrant, and cos function has negative values in second quadrant.

[outer briar]

@gleaming owl oh how careless i am! Thankyou! Haven't read the question properly, my bad!

[hushed sphinx]

There are (6 choose 2) ways to pick to two men to go on the committee, but for each of those choices there's more than one possibility for who the four women will be.

[swift flume]

right

man 1 isnt the same as man 2

[full pagoda]

you have to be at least 13 to use discord

[hushed sphinx]

Doesn't matter what you think. American law does not allow Discord the company to provide a service to under-13s without particular expensive and cumbersome protection measures.

[stuck lark]

@hushed sphinx @full pagoda for future reference pls report underage ppl to mods

[fleet grove]

wondering how that doesn't end up as √ 109

that's supposed to be formula

so wouldn't it end up as √ (-3)^2 + (-10)^2

which just simplifies to √109

<@&286206848099549185>

[uncut mulch]

where's (-10)^2 coming from

[vivid void]

don’t immediately ping helpers

there are solutions to that

no

[uncut mulch]

where are you getting the idea that there are no solutions

what exactly did you try

and what exactly did you put into (ew) mathway that's supposedly giving no solutions

[fleet grove]

I'm dumb I accidentally added the negatives instead of subtracting

[uncut mulch]

wdym by

then made it into
quad equation

is not a quadratic equation

[old isle]

getting a 100% by cheating is no fun tho

[uncut mulch]

you're equation can't be expressed in the form
ax^2+bx+c=0
(where a,b,c are constants, a != 0)

there aren't nice solutions to this equation

you could consider squaring both sides of

x^2-9=sqrtx
you'd end up with a quartic equation
however you'd probably need to apply methods of approximation (newtons, midpoint)

and/or graph it with compute assistance for approximate solutions

[old isle]

is the question x^2-9=√x?

what did u get

[uncut mulch]

is the question x^2-9=√x?
that's not what was initially provided

[old isle]

ohh

x^2-6x+9=√x?

ohh

np

[viscid thistle]

is there an analytic way to calculate the "a" value at which these two graphs touch eachother just once?

[old isle]

find the derivative of both graphs

equate them to each other

i think

[viscid thistle]

... This is the pre calculus chat

Treat it like a system of equations and solve a^x = log_a(x)

hmm im not actauly sure itas possible yto solve that one

hmm well

those functions are inverses, and an inverse is just a flip across the y=x function, so any point they intersect would also intersect the y=x function.

so the answer is the same as the system of equations
y = x
y = a^x
giving us a^x = x
which is not analytically solvable

looks like something you would want the Lambert W function for

that is a very intresting value for a you got

do you know that the equations are equal at that specific value of a?

OHOHOHOH i miss read it, your looking for A

in which case no i dont know of any algebraic method

[viscid thistle]

hold on

@viscid thistle i think it might be

$\sqrt[e]{e}$

[obsidian monolith]

aspwil

[viscid thistle]

or about 1.44466786101

[old isle]

how'd u solve that

[viscid thistle]

thats a very good question

unfonotily im not quite sure if/why the mathematical fuckery i just pulled worked

[old isle]

insane

[viscid thistle]

heres what i did, it seems to work but again i have no idea if it actauly did or exactly why
so its the system of equations where
y = x
y = a^x
so x = a^x via system of equations
x^(1/x) = a
looking at its graph it has 1 intresting point where its at a hump (see picture)
solving f'(x) = 0 give us the x value of that point, and the y value is e^(1/e)
which seems like it might be correct

its on hella shaky ground, and comptly unproven by logic, but its better then nothing?

so the point is maybe (e^(1/e), e^(1/e))

but that a very big maybe

[old isle]

fundamentally speaking, i dont think you can find the value of a solely by algebra

you'd have to assume values of a

and keep verifying it with the equation

[viscid thistle]

well e is not part of algebra its a transendental number

so its definitly not an obtainable number through algebraic means

[viscid thistle]

another (unverified) intresting thing about e^(1/e)

huh and that point of intersection looks like it migh be (e,e)

huh that seems to check out

in fact there some real weirdness going on

the intsection points of where
y = a^x
and
y = log_a(x)
seem to be the x values of the intersections of
y = x^(1/x)
and
y = a

wait that makes sence

the inverse of a function is just a flip along the line y=x
thus if a function equals its inverse it the soltion is the same as the intersection with y=x
so the intersections of y = a^x and y = log_a(x) is the same as
y= a^x and y = x
and these intersections are the answer to the equation x = a^x
solving for a gives us a = x^(1/x)
now we just need to find the first value of a that goes from giving us no points
(like so see no intersection for a = 1.6)

as a gets smaller (1.5 in this pic) the lines get closer

we need to decreese a untill we get a working point

a = x^(1/x) this is the same as choosing a value for a in this.

so we need to bring a down until we see a value at which we have an answer for this, sop the first point we see will be the highest value

so were just finding the highest a value in which a = x^(1/x) has a solution

(which we dont have a way to do in algebra, but we do in calculus)

setting the derivitive with respect to x to 0 and solving will yield the highest value of a for which a = x^(1/x)

solving for the derivitaive using symbolab give us this

solving for when the derivitive equals 0 give us this

which mean the x value is equal to e

plugging that back into a = x^(1/x) we get a = e^(1/e)

and there have the answer

so the logic is 100% sound (as far as i can tell), just not a possibilty to solve using just algebra

[viscid thistle]

wow this could make a realy cool math video i think i might make one based off of it

[viscid thistle]

yeah i could totaly explain this as a like 5 or 6 min long video, vissuals for what goping on would make it so much better

[old isle]

wow u figured out using derivatives

precalc chat

[viscid thistle]

can someone explain why cos dosent change to sin here?

and why we only derivate the x2

[trim hemlock]

that's because you are differentiating wrt x, cos(pi/4) is a constant

[viscid thistle]

ohhh i see then

[mint cedar]

precalculus khan academy good?

[old isle]

yes

[steel fern]

im looking to find some simulations that I can leverage precalc and algebra with. I want to program something that uses math. in order to have fun doing math. does anyone have any ideas of something with only a strong algebra backround

[viscid thistle]

Is my solution right or wrong someone please tell me

[viscid thistle]

Hello rn we're studying formulas like U/V = U'V - V'U / V^2, the chapter is called derivatives (as my professor said), but when I google it, I get nothing like these formulas, last time I checked I got something like d/dx something, not only different variable but different formula completely, what should I search to find the chapter that I am studying RN

https://www.cuemath.com/calculus/quotient-rule/

[finite heath]

^ that's that you just said but more formal

[worthy frost]

memorize this

[viscid thistle]

@tame lagoon Do you know the definition of a left-hand and right-hand limit?

Ok cool. And so how would you calculate g(-4.01)?

Because you are confused about the definition of g?

If f(x) = sqrt(4 - 3x), then what is f(-4.01)?

With f, yes just make the substitution.

But there is some nuance with your original function g.

I'm trying to explain.

Well actually, I'm trying to guess what you are confused about.

You said that you don't know how to calculate g(-4.01), but it's not clear to me exactly what you don't understand about it.

So explain what you did for (a)

Ok cool.

And what about for (b)?

Alright, so I don't think that reasoning is quite right. Think about (a) some more.

You substituted -4 for x in g right?