Question

How do you write $ - 4 + 8i $ in polar form?

Answer 1

Hint: In order to obtain the equivalent polar form of the given complex number, obtain the values of variables $ a,b $ by comparing the number with $ z = a + bi $ . The polar form of any complex number $ z = a + bi $ is equal to $ z = r\left( {\cos \theta + i\sin \theta } \right) $ . Find out the value of $ r $ by using $ r = \left| z \right| = \sqrt {{a^2} + {b^2}} $ and define the value of $ \theta $ by using $ \theta = {\tan ^{ - 1}}\left( {\dfrac{b}{a}} \right) $ . And put all the values back in $ z = r\left( {\cos \theta + i\sin \theta } \right) $ to obtain the required result.
Formula:
 $ {i^2} = - 1 $
 $ {a^m}.{a^n} = {a^{m + n}} $

Complete step by step solution:
We are Given a complex number $ - 4 + 8i $ let it be z
 $ z = - 4 + 8i $ --(1)
Here i is the imaginary number which is commonly known as the iota.
The form which we are given is called the rectangular form of complex numbers.
The polar form of any complex number $ z = a + bi $ is equal to $ z = r\left( {\cos \theta + i\sin \theta } \right) $
In order to convert the given complex number into the polar form, compare it with $ z = a + bi $ to obtain the values of the variables. We get
 $ a = - 4\,and\,b = 8 $
Now calculating the value of $ r $ which is equal to
  $ r = \left| z \right| = \sqrt {{a^2} + {b^2}} $
Putting the values of variables
\[
  r = \sqrt {{{\left( { - 4} \right)}^2} + {{\left( 8 \right)}^2}} \\
   = \sqrt {80} \\
   = 4\sqrt 5 \;
 \]
Now finding the value of $ \theta $ as
 $ \theta = {\tan ^{ - 1}}\left( {\dfrac{b}{a}} \right) $
 $
  \theta = {\tan ^{ - 1}}\left( {\dfrac{8}{{ - 4}}} \right) \\
  \theta = {\tan ^{ - 1}}\left( { - 2} \right) \\
  \theta = {\text{ - 0}}{\text{.35241638}}\pi \;
  $
So , our polar form will be
 $
  z = r\left( {\cos \theta + i\sin \theta } \right) \\
  z = 4\sqrt 5 \left( {\cos \left( {{\text{ - 0}}{\text{.35241638}}\pi } \right) + \sin \left( {{\text{ - 0}}{\text{.35241638}}\pi } \right)i} \right) \;
  $
As we know rule of trigonometry that $ \cos \left( { - x} \right) = \cos x $ and $ \sin \left( { - x} \right) = - \sin x $
 $ z = 4\sqrt 5 \left( {\cos \left( {{\text{0}}{\text{.35241638}}\pi } \right) - \sin \left( {{\text{0}}{\text{.35241638}}\pi } \right)i} \right) $
Therefore, the required answer is $ z = 4\sqrt 5 \left( {\cos \left( {{\text{0}}{\text{.35241638}}\pi } \right) - \sin \left( {{\text{0}}{\text{.35241638}}\pi } \right)i} \right) $ .

Note: 1. Complex numbers are very useful in representing periodic motion like water waves, light waves and current and many more things which depend on sine or cosine waves.
2. Complex conjugate of $ a + ib $ is $ a - ib $
3. $ {i^3} $ is equal to $ - i $ as $ {i^3} = i.{i^2} = i.\left( { - 1} \right) = - i $