Cartesian Formulaes

Formula Sheet for Complex Numbers

1. Cartesian to Polar Conversion

Form:
$z=x+jy$
Polar Form:
$z=r{e}^{j\theta }$

Steps:

1. Magnitude $r$:

   $$r=|z|=\sqrt{x^2+y^2}$$

2. Angle $\theta$:

   $$\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$$
   • Adjust $\theta$ based on the quadrant:
     • Quadrant I: $\theta$ as calculated.
     • Quadrant II: $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)+\pi$
     • Quadrant III: $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)-\pi$
     • Quadrant IV: $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

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2. Polar to Cartesian Conversion

Form:
$z=r{e}^{j\theta }$

Steps:

1. Real Part $x$:

   $$x=r\cos (\theta )$$

2. Imaginary Part $y$:

   $$y=r\sin (\theta )$$

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3. Examples for Practice

1. Convert the following to polar form:
   • $5$
   • $-2$
   • $-3j$
   • $\frac{1}{2}-j\frac{\sqrt{3}}{2}$
   • $1+j$
   • $j(1-j{)}^2$
   • $\frac{(1+j)}{(1-j)}$
   • $\frac{(\sqrt{2}+j\sqrt{2})}{(1+j\sqrt{3})}$

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Notes

• Ensure to consider the range of $\theta$ to be $-\pi <\theta \le \pi$.