easy problem

[vague chasm]

easy problem for timmy
$\sum_{n=0}^{\infty} \frac{\sin(n\theta)}{2^n}
Where $\cos(\theta)=\frac{1}{5}$

[blazing sky]

;( | 追放された興奮
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[carmine sparrow]

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[rocky ore]

@rigid lintel

[rigid lintel]

Um

What am I trying to find

[vague chasm]

the sum

start with finding $\sin(\theta)$

[blazing sky]

;( | 追放された興奮

[plucky bone]

what is arccos 1/5

search that

[rigid lintel]

Ok so since costheta =1/5

That means sintheta =

Idk

[vague chasm]

what is cos(theta)

[plucky bone]

arccos 1/5 about 78.46°

[rigid lintel]

1/5?

[plucky bone]

so sin(78.46°)

[vague chasm]

$\cos(\theta)=\sqrt{1-sin^2(\theta)}$

[blazing sky]

;( | 追放された興奮

[vague chasm]

no.

we're looking for exact values here

[plucky bone]

ok

[rigid lintel]

1/25=1-sin^2

[vague chasm]

no

[rigid lintel]

24/25=sin2

[vague chasm]

yes

[plucky bone]

wai

[rigid lintel]

Sqrt24/5

[vague chasm]

👍

okay now that we know that

we need to use de moivre's theorem

[rigid lintel]

Ah yes my favourite

[plucky bone]

2ⁿ which is below will definitely converge to 1, but what about the one above?

[vague chasm]

which states that $\cos(n\theta)+i\cdot\sin(\theta)=[\cos(\theta)+i\cdot\sin(\theta)]^n$

[blazing sky]

;( | 追放された興奮

[rigid lintel]

Obviously

[vague chasm]

so now we use this

[rigid lintel]

So we sub in the values for sin and cos I assume

[plucky bone]

but the question there is i

[vague chasm]

and therefore $\mathrm{Im}[(\cos(\theta)+i\cdot\sin(\theta))^n]=\sin(n\theta)$

[rigid lintel]

What does that Re mean

[vague chasm]

Im means the imaginary part of

[rigid lintel]

Okay

[blazing sky]

;( | 追放された興奮

[rigid lintel]

Wait

[vague chasm]

so even better

?

[rigid lintel]

Sin(ntheta) is imaginary?

[vague chasm]

its the imaginary part of the complex number

so yes

[rigid lintel]

Ok I'm lost

Are you gaslighting me?

[vague chasm]

anyways, we can thus say that $\mathrm{Im}[(e^{i\theta})^n}]=\sin(n\theta)$

[vague chasm]

no, this is a perfectly reasonable problem.

[plucky bone]

sin(n) where n = real output real num

[blazing sky]

;( | 追放された興奮
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[rigid lintel]

EULERS IDENTITY

[vague chasm]

yeah

[vague chasm]

from here we know that $\mathrm{Im}(e^{i\cdot{n}\theta})=\sin(n\theta)$

[blazing sky]

;( | 追放された興奮
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[rigid lintel]

Did I mean to write like that?

You*

[vague chasm]

no

i need to fix it

$\mathrm{Im}(e^{i\cdot{n}\theta})=\sin(n\theta)$

[blazing sky]

;( | 追放された興奮

[vague chasm]

and basically we have $\sum_{n=0}^{\infty} \frac{\mathrm{Im}(e^{i\cdot n\theta})}{2^n}$

[blazing sky]

;( | 追放された興奮

[rigid lintel]

Sorry I got some stuff to do you might have to get someone else to help with the question

[vague chasm]

which is $\mathrm{Im}\left(\sum_{n=0}^{\infty} \left(\frac{e^{i\theta}}{2}\right)^n\right)$

[blazing sky]

;( | 追放された興奮

[rigid lintel]

Geometric series!

[vague chasm]

👍

[wide jungle]

@vague chasm

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[vague chasm]

+close

[wide jungle]

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