Home » Blog » Convert complex number in polar form to the form a + bi

Convert complex number in polar form to the form a + bi

by RoRi

Convert each of the following complex numbers given in polar form to the form a+bi.

  1. e^{\frac{\pi i}{2}}.
  2. 2e^{-\frac{\pi i}{2}}.
  3. 3e^{\pi i}.
  4. -e^{-\pi i}.
  5. i + e^{2 \pi i}.
  6. e^{\frac{\pi i}{4}}.
  7. e^{\frac{\pi i}{4}} - e^{-\frac{\pi i}{4}}.
  8. \displaystyle{ \frac{1 - e^{\frac{\pi i}{2}}}{1 + e^{\frac{ \pi i}{2}}}}.
  1. Using the definition of the complex exponential (e^{i \theta} = \cos \theta + i \sin \theta) we have

        \[ e^{ \frac{\pi i}{2}} = \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} = 0 + i = i. \]

  2. Again, using the definition of the complex exponential we have,

        \[ 2e^{-\frac{\pi i}{2}} = 2 \left(\cos \frac{-\pi}{2} + i \sin \frac{-\pi}{2} \right)= -2i. \]

  3. We have

        \[ 3e^{\pi i} = 3 \left( \cos \pi + i \sin \pi \right) = -3. \]

  4. We have

        \[ -e^{-\pi i} = - \left( \cos (-\pi) + i \sin (-\pi) \right) = 1. \]

  5. We have

        \[ i + e^{2 \pi i} = i + \cos (2 \pi) + i \sin (2 \pi) = i + 1. \]

  6. We have

        \[ e^{ \frac{\pi i}{4}} = \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} = \frac{1 +i}{\sqrt{2}}. \]

  7. We have

        \[ e^{\frac{\pi i}{4}} - e^{-\frac{\pi i}{4}} = \frac{1+i}{\sqrt{2}} - \frac{1-i}{\sqrt{2}} = \sqrt{2} i. \]

  8. We have

        \[ \frac{1 - e^{\frac{\pi i}{2}}}{1 + e^{\frac{\pi i}{2}}} = \frac{1-i}{1+i} = \frac{-2i}{2} = -i. \]

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):