6.5: The Trigonometric Form of Complex Numbers
Learning Objectives
 Understand the relationship between the rectangular form of complex numbers and their corresponding polar form.
 Convert complex numbers from standard form to polar form and vice versa.
A number in the form
The Trigonometric or Polar Form of a Complex Number
The following diagram will introduce you to the relationship between complex numbers and polar coordinates.
In the figure above, the point that represents the number
If we apply the first two equations to the point
The right side of this equation
It is now time to implement these equations perform the operation of converting complex numbers in standard form to complex numbers in polar form. You will use the above equations to do this.
Example 1: Represent the complex number
Solution: As discussed in the Prerequisite Chapter, here is the graph of
Converting to polar from rectangular,
So, the polar form is
Another widely used notation for the polar form of a complex number is
Example 2: Express the following polar form of each complex number using the shorthand representations.
a)
b)
Solution:
a)
b)
Example 3: Represent the complex number
Solution: From the rectangular form of
This is the reference angle so now we must determine the measure of the angle in the third quadrant.
One polar notation of the point
So far we have expressed all values of theta in degrees. Polar form of a complex number can also have theta expressed in radian measure. This would be beneficial when plotting the polar form of complex numbers in the polar plane.
The answer to the above example
Now that we have explored the polar form of complex numbers and the steps for performing these conversions, we will look at an example in circuit analysis that requires a complex number given in polar form to be expressed in standard form.
Example 4: The impedance
Solution: The value for \begin{align*}Z\end{align*}
\begin{align*}Z &= 4650 (\cos(35.2^\circ) + i \sin(35.2^\circ))\\ x &= 4650 \cos(35.2^\circ) \rightarrow 3800\\ y &= 4650 \sin (35.2^\circ) \rightarrow 2680\end{align*}
Therefore the standard form is \begin{align*}Z= 3800  2680i\end{align*}
Points to Consider
 Is it possible to perform basic operations on complex numbers in polar form?
 If operations can be performed, do the processes change for polar form or remain the same as for standard form?
Review Questions
 Express the following polar forms of complex numbers in the two other possible ways.

\begin{align*}5 \ cis \frac{\pi}{6}\end{align*}
5 cisπ6 
\begin{align*}3 \angle 135^\circ\end{align*}
3∠135∘  \begin{align*}2 \left(\cos \frac{2\pi}{3}+i \sin \frac{2\pi}{3}\right)\end{align*}

\begin{align*}5 \ cis \frac{\pi}{6}\end{align*}
 Express the complex number \begin{align*}6  8i\end{align*} graphically and write it in its polar form.
 Express the following complex numbers in their polar form.
 \begin{align*}4 + 3i\end{align*}
 \begin{align*}2 + 9i\end{align*}
 \begin{align*}7  i\end{align*}
 \begin{align*}5  2i\end{align*}
 Graph the complex number \begin{align*}3(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4})\end{align*} and express it in standard form.
 Find the standard form of each of the complex numbers below.
 \begin{align*}2 \ cis \frac{\pi}{2}\end{align*}
 \begin{align*}4 \angle \frac{5\pi}{6}\end{align*}
 \begin{align*}8 \left( \cos \left(\frac{\pi}{3}\right)+i \sin \left(\frac{\pi}{3}\right)\right)\end{align*}
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Date Created:
Feb 23, 2012Last Modified:
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