comp math

[spiral hornet]

Here are the two questions

1. Find $\Im(\cos(12^{\circ})+i\sin(12^{\circ})+\cos(48^{\circ})+i\sin(48^{\circ}))^6$
2. If $\theta$ is a constant such that $0<\theta<\pi$ and $x+\frac{1}{x}=2\cos(\theta)$, then for each positive integer $n$, find $x^n+\frac{1}{x^n}$ in terms of $n$ and $\theta$
   Number 1 i just don't understand how to do, number 2 i got a solution but idk if its right so i want the helper to work me through it

[neon mulch]

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[quaint junco]

flying green people eater

[modern dock]

i suppose you could send your solution to 2.

also, there seems to be some missing brackets in the first question

[spiral hornet]

(im too lazy to fix it)

[modern dock]

ok

for the first question, i would calculate cis(12) + cis(48)

you may place the numbers in a triangle or treat them as vectors

[spiral hornet]

see i tried that but idk how

[modern dock]

could you draw an argand diagram and plot cis(12) and cis(48) on the diagram?

[spiral hornet]

probably but im gonna leave for school

[modern dock]

then create the point A that forms a parallelogram with one diagonal from cis(12) to cis(48) and the other from 0 to A

[spiral hornet]

oh

wait that makes sense lol

[modern dock]

that is cis(12) + cis(48)

[spiral hornet]

🤦‍♂️

i shoulda used vectors bru

[modern dock]

the magnitude may require trigonometry to find, but the argument will be really quite nice

[spiral hornet]

so ya i will do that when i get paper

[modern dock]

especially multiplied by 6

[spiral hornet]

seems simple enough

[modern dock]

yes

[spiral hornet]

for 2

my solution is $2\cos(n\theta)$

[quaint junco]

flying green people eater

[spiral hornet]

i solved for x and did a little exponential form shennanigans

can you verify?

[spiral hornet]

$\int_{-r}^{r} \sqrt{r^2-x^2} dx$

[quaint junco]

flying green people eater

[spiral hornet]

let $x=r\sin(\theta)$, then $dx=r\cos(\theta)$

when $x=r$, $\theta=\frac{\pi}{2}$ and when $x=-r$, $\theta=\frac{3\pi}{2}$

[quaint junco]

flying green people eater

flying green people eater

[spiral hornet]

so we have $2r^2\int_{-\pi/2}^{\pi/2} \cos^{2}(\theta) d\theta$

and the applications are obvious from here

[quaint junco]

flying green people eater

[spiral hornet]

,w $2r^2\int_{-\pi/2}^{\pi/2} \cos^{2}(\theta) d\theta$

[quaint junco]

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[modern dock]

@spiral hornet |x + 1/x| ≥ 2 > |2cos(x)|; are you certain that the question is solveable?

[spiral hornet]

yes i checked solution book

it’s from AOPS Problem Solving Volume 2 for referencr

[spiral hornet]

If $f(z)=a_nz^n+a_{n-1}z^{n-1}+...a_0$ and $0\leq a_k\leq 1$ show that $z^{n+1}=1$

[quaint junco]

flying green people eater

[modern dock]

is x real?

[spiral hornet]

idk about that

[modern dock]

send screenshot

[spiral hornet]

if i take a picture it’s practically unreadable

i scribbled all over it

[modern dock]

that is your problem for writing on the book

send it

[spiral hornet]

alr when i get home

[spiral hornet]

uh

do you want me to send the solution or the problem

[modern dock]

problem

[spiral hornet]

okay

sorry i was doing meaningless coding

[spiral frigate]

@spiral hornet

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[spiral hornet]

+close

[spiral frigate]

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[spiral frigate]

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