Change In To Polar Form
Memorize The Following Diagram To Convert Any Cartesian Form To Polar Form Within Seconds
Cartesian To Polar
Easiest Way To Convert Cartesian Form In To Polar Form. A Complex Number In Cartesian Form Can Be Any One Of The 8 Possible Places As Mentioned In The Above Diagram
Convert \( Z = -\frac{1}{2} + i \frac{\sqrt{3}}{2} \) into modulus–argument (polar) form
\[ \]
Step 1: Find \( r \)**
\[
r = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2}
\]
\[
r = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1
\]
Step 2: Identify the position of the complex number
\[
\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \;\; \Rightarrow \;\; (-,+) \;\; \text{lies in Quadrant II}
\]
Step 3: Find the argument
\[
\Theta = \pi - \tan^{-1}\left|\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}}\right|
\]
\[
\Theta = \pi - \tan^{-1}(\sqrt{3})
\]
\[
\Theta = \pi - \frac{\pi}{3}
\]
\[
\Theta = \frac{2\pi}{3}
\]
Step 4: Write in polar form
\[
-\frac{1}{2} + i \frac{\sqrt{3}}{2}
= 1 \Big( \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} \Big)
\]
Special Case : The Complex Number Lies On The Axis
Convert the complex number \( z = -3i \) into polar form
\[ z = -3i \]
\[ z = 0 + (-3)i \;\;\;\; \Rightarrow \;\; (0, -3) \]
\[ r = \sqrt{0^2 + (-3)^2} = 3 \]
\[ z = -3i = (0, -3) \;\; \text{is on the negative y-axis} \]
From the picture: \(\; \Theta = -\frac{\pi}{2} \)
\[ \text{Therefore,} \;\; -3i = 3 \Big( \cos \big(-\frac{\pi}{2}\big) + i \sin \big(-\frac{\pi}{2}\big) \Big) \]