# Rectangular form

#### Petrus

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

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

Regards,

#### Klaas van Aarsen

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

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

Regards,

Hey Petrus!

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

#### Petrus

##### Well-known member
Hey Petrus!

What is $$\displaystyle \frac {100\pi}{3} \pmod{2\pi}$$?
$$\displaystyle \frac{4}{3}$$

- - - Updated - - -

Thanks I like Serena I see what I did wrong Regards,

#### Klaas van Aarsen

##### MHB Seeker
Staff member
$$\displaystyle \frac{4}{3}$$
That should be $$\displaystyle \frac{4\pi}{3}$$. (Yeah, I know, I'm a nitpicker.)

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

#### Petrus

##### Well-known member
(Yeah, I know, I'm a nitpicker
nitpicker or not, I am grateful for the fast responed!

Regards,

#### MarkFL

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