What is the Result of Raising a Complex Number to a Power?

[Petrus]

Hello MHB,
calculate $$\left(\frac{1}{2}+i\frac{\sqrt{3}}{2} \right)^{100}$$ in the form $$a+ib$$

progress:
I start to calculate argument and get it to $$r=1$$ (argument)
then $$\cos\theta=\frac{1}{2} \ sin\theta=\frac{\sqrt{3}}{2}$$ we se it's in first quadrant( where x and y is positive)
$$1*e^{i\frac{100\pi}{3}}$$
notice that we can always take away 2pi so we can simplify that to
$$1*e^{i\frac{\pi}{3}}$$
$$1*e^{i\frac{\pi}{3}}=\cos(\frac{\pi}{3}) + i \sin (\frac{\pi}{3}) = \frac{1}{2}+i\frac{\sqrt{3}}{2}$$
but the facit says $$\frac{-1}{2}-i\frac{\sqrt{3}}{2}$$

Regards,

 

[I like Serena]

Hello MHB,
calculate $$\left(\frac{1}{2}+i\frac{\sqrt{3}}{2} \right)^{100}$$ in the form $$a+ib$$

progress:
I start to calculate argument and get it to $$r=1$$ (argument)
then $$\cos\theta=\frac{1}{2} \ sin\theta=\frac{\sqrt{3}}{2}$$ we se it's in first quadrant( where x and y is positive)
$$1*e^{i\frac{100\pi}{3}}$$
notice that we can always take away 2pi so we can simplify that to
$$1*e^{i\frac{\pi}{3}}$$
$$1*e^{i\frac{\pi}{3}}=\cos(\frac{\pi}{3}) + i \sin (\frac{\pi}{3}) = \frac{1}{2}+i\frac{\sqrt{3}}{2}$$
but the facit says $$\frac{-1}{2}-i\frac{\sqrt{3}}{2}$$

Regards,

Hey Petrus!

What is $$\frac {100\pi}{3} \pmod{2\pi}$$?

 

[Petrus]

Hey Petrus!

What is $$\frac {100\pi}{3} \pmod{2\pi}$$?

$$\frac{4}{3}$$

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Thanks I like Serena I see what I did wrong :)

Regards,

 

[I like Serena]

$$\frac{4}{3}$$

That should be $$\frac{4\pi}{3}$$. (Yeah, I know, I'm a nitpicker.)

So what's $$\cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3}$$?

 

[Petrus]

(Yeah, I know, I'm a nitpicker

nitpicker or not, I am grateful for the fast responed!

Regards,

 

[MarkFL]

If I were to work the problem, I would write:

$$\left(\frac{1}{2}+i\frac{\sqrt{3}}{2} \right)^{100}= \left(\cos\left(\frac{\pi}{3}+2k\pi \right)+i\sin\left(\frac{\pi}{3}+2k\pi \right) \right)^{100}$$

Now applying de Moivre's theorem we have:

$$\cos\left(\frac{100\pi}{3}+200k\pi \right)+i\sin\left(\frac{100\pi}{3}+200k\pi \right)=\cos\left(\frac{4\pi}{3}+232k\pi \right)+i\sin\left(\frac{4\pi}{3}+232k\pi \right)= -\left(\frac{1}{2}+i\frac{\sqrt{3}}{2} \right)$$