Complex Number Euler's Law addition, Finding the real and imaginary parts

Question

Euler's Law addition, Finding the real and imaginary parts of $\ z_1^2+ jz_2 $

Where $\ z_1=e^{j\pi/4} $ and $\ z_2=e^{-j\pi/3} $

I don't really understand this question, but I did some working:

$\ z_1^2 = e^{\pi/2} $

$\ z_1^2+jz_2 = e^{j\pi/2}+e^{-j\pi/2} $

$\ = 2\cos{\pi/2-\pi/3} = \sqrt{3} $

So I don't know what to do now.

Answer 1

HINT generalize the problem, when $\text{z}_1\space\wedge\space\text{z}_2\in\mathbb{C}$:

$$\text{Q}=\text{z}_1^2+\text{z}_2i=\left(\Re[\text{z}_1]+\Im[\text{z}_1]i\right)^2+\left(\Re[\text{z}_2]+\Im[\text{z}_2]i\right)i$$

Now, we can set:

1. Using $i^2=-1$: $$\left(\Re[\text{z}_2]+\Im[\text{z}_2]i\right)i=\Re[\text{z}_2]i+\Im[\text{z}_2]i^2=-\Im[\text{z}_2]+\Re[\text{z}_2]i$$
2. $$\left(\Re[\text{z}_1]+\Im[\text{z}_1]i\right)^2=\left(\Re[\text{z}_1]+\Im[\text{z}_1]i\right)\left(\Re[\text{z}_1]+\Im[\text{z}_1]i\right)=\Re^2[\text{z}_1]-\Im^2[\text{z}_1]+2\Re[\text{z}_1]\Im[\text{z}_1]i$$

So, for $\text{Q}$ we know:

• $$\Re\left[\text{Q}\right]=\Re\left[\text{z}_1^2+\text{z}_2i\right]=\Re^2[\text{z}_1]-\Im^2[\text{z}_1]$$

• $$\Im\left[\text{Q}\right]=\Im\left[\text{z}_1^2+\text{z}_2i\right]=2\Re[\text{z}_1]\Im[\text{z}_1]$$

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Now, using Euler's formula we can write $\text{z}$ when $\text{z}\in\mathbb{C}$:

$$\text{z}=\Re[\text{z}]+\Im[\text{z}]i=|\text{z}|e^{\left(\arg(\text{z})+2\pi k\right)i}=|\text{z}|\cos(\arg(\text{z})+2\pi k)+|\text{z}|\sin(\arg(\text{z})+2\pi k)i$$

Where $|\text{z}|=\sqrt{\Re^2[\text{z}]+\Im^2[\text{z}]}$, $\arg(\text{z})$ is the complex argument of $\text{z}$ and $k\in\mathbb{Z}$.

Answer 2

The problem is to reduce your number $w=z_1^2+iz_2$ to be of the form $x+iy$, where $x,y\in\mathbb{R}$. We call $x$ the real part and $y$ the imaginary part. We have

\begin{align} w&=z_1^2+iz_2 \\ z_1&=e^{i\pi/4} \\ z_2&=e^{-i\pi/3} \end{align}

which gives

\begin{align} w&=e^{i\pi/2}+ie^{-i\pi/3} \\ &=i+ie^{-i\pi/3} \\ &=i+i\left(\frac{1-i\sqrt{3}}{2} \right)\\ &=\frac{\sqrt{3}}{2}+i\frac{3}{2} \end{align}

Your attempt contains multiple errors and suggests that you should work on your algebra skills (reducing expressions, exponentiation rules, etc.).