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6.5: The Trigonometric Form of Complex Numbers

Difficulty Level: At Grade Created by: CK-12
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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 \begin{align*}a + bi\end{align*}a+bi, where \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are real numbers, and \begin{align*}i\end{align*} is the imaginary unit, or \begin{align*}\sqrt{-1}\end{align*}, is called a complex number. Despite their names, complex numbers and imaginary numbers have very real and significant applications in both mathematics and in the real world. Complex numbers are useful in pure mathematics, providing a more consistent and flexible number system that helps solve algebra and calculus problems. We will see some of these applications in the examples throughout this lesson.

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 \begin{align*}x + yi\end{align*} was plotted and a vector was drawn from the origin to this point. As a result, an angle in standard position, \begin{align*}\theta\end{align*}, has been formed. In addition to this, the point that represents \begin{align*}x + yi\end{align*} is \begin{align*}r\end{align*} units from the origin. Therefore, any point in the complex plane can be found if the angle \begin{align*}\theta\end{align*} and the \begin{align*}r-\end{align*} value are known. The following equations relate \begin{align*}x, y, r\end{align*} and \begin{align*}\theta\end{align*}.

\begin{align*}x=r \cos \theta && y=r \sin \theta && r^2=x^2+y^2 && \tan \theta=\frac{y}{x}\end{align*}

If we apply the first two equations to the point \begin{align*}x + yi\end{align*} the result would be:

\begin{align*}x + yi = r \cos \theta + r i \sin \theta \rightarrow r (\cos \theta + i \sin \theta)\end{align*}

The right side of this equation \begin{align*}r(\cos \theta + i \sin \theta)\end{align*} is called the polar or trigonometric form of a complex number. A shortened version of this polar form is written as \begin{align*}r \ cis \ \theta\end{align*}. The length \begin{align*}r\end{align*} is called the absolute value or the modulus, and the angle \begin{align*}\theta\end{align*} is called the argument of the complex number. Therefore, the following equations define the polar form of a complex number:

\begin{align*}r^2=x^2+y^2 && \tan \theta =\frac{y}{x} && x+yi=r(\cos \theta + i \sin \theta)\end{align*}

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 \begin{align*}5 + 7i\end{align*} graphically and express it in its polar form.

Solution: As discussed in the Prerequisite Chapter, here is the graph of \begin{align*}5 + 7i\end{align*}.

Converting to polar from rectangular, \begin{align*}x = 5\end{align*} and \begin{align*}y = 7\end{align*}.

\begin{align*}& r=\sqrt{5^2+7^2}=8.6 && \tan \theta=\frac{7}{5}\\ &&& \tan ^{-1}(\tan \theta)=\tan ^{-1}\frac{7}{5}\\ &&& \theta=54.5^\circ\end{align*}

So, the polar form is \begin{align*}8.6(\cos 54.5^\circ + i \sin 54.5^\circ)\end{align*}.

Another widely used notation for the polar form of a complex number is \begin{align*}r \angle \theta=r (\cos \theta + i \sin \theta)\end{align*}. Now there are three ways to write the polar form of a complex number.

\begin{align*}x+yi=r(\cos \theta+i \sin \theta) && x+yi=rcis \theta && x+yi=r \angle \theta\end{align*}

Example 2: Express the following polar form of each complex number using the shorthand representations.

a) \begin{align*}4.92 (\cos 214.6^\circ + i \sin 214.6^\circ)\end{align*}

b) \begin{align*}15.6 (\cos 37^\circ + i \sin 37^\circ)\end{align*}


a) \begin{align*}4.92 \angle 214.6^\circ\end{align*}

\begin{align*}4.92 \ cis \ 214.6^\circ\end{align*}

b) \begin{align*}15.6 \angle 37^\circ\end{align*}

\begin{align*}15.6 \ cis \ 37^\circ\end{align*}

Example 3: Represent the complex number \begin{align*}-3.12 - 4.64i\end{align*} graphically and give two notations of its polar form.

Solution: From the rectangular form of \begin{align*}-3.12 - 4.64i \ x = - 3.12\end{align*} and \begin{align*}y = - 4.64\end{align*}

\begin{align*}r &= \sqrt{x^2+y^2}\\ r &= \sqrt{(-3.12)^2+(-4.64)^2}\\ r &= 5.59\end{align*}

\begin{align*}\tan \theta &= \frac{y}{x}\\ \tan \theta &=\frac{-4.64}{-3.12}\\ \theta&=56.1^\circ\end{align*}

This is the reference angle so now we must determine the measure of the angle in the third quadrant. \begin{align*}56.1^\circ + 180^\circ = 236.1^\circ\end{align*}

One polar notation of the point \begin{align*}-3.12 - 4.64i\end{align*} is \begin{align*}5.59\end{align*} \begin{align*}(\cos 236.1^\circ + i \sin 236.1^\circ)\end{align*}. Another polar notation of the point is \begin{align*}5.59 \angle 236.1^\circ\end{align*}

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 \begin{align*}-3.12 - 4.64i\end{align*} with theta expressed in radian measure would be:

\begin{align*}& \tan \theta =\frac{-4.64}{-3.12} && \tan \theta=.9788(\text{reference angle})\\ &&& 0.9788+3.14=4.12 \ \text{rad}.\\ & 5.59(\cos 4.12+i \sin 4.12)\end{align*}

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 \begin{align*}Z\end{align*}, in ohms, in an alternating circuit is given by \begin{align*}Z=4650 \angle -35.2^\circ\end{align*}. Express the value for \begin{align*}Z\end{align*} in standard form. (In electricity, negative angles are often used.)

Solution: The value for \begin{align*}Z\end{align*} is given in polar form. From this notation, we know that \begin{align*}r = 4650\end{align*} and \begin{align*}\theta = -35.2^\circ\end{align*} Using these values, we can write:

\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*} ohms.

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

  1. Express the following polar forms of complex numbers in the two other possible ways.
    1. \begin{align*}5 \ cis \frac{\pi}{6}\end{align*}
    2. \begin{align*}3 \angle 135^\circ\end{align*}
    3. \begin{align*}2 \left(\cos \frac{2\pi}{3}+i \sin \frac{2\pi}{3}\right)\end{align*}
  2. Express the complex number \begin{align*}6 - 8i\end{align*} graphically and write it in its polar form.
  3. Express the following complex numbers in their polar form.
    1. \begin{align*}4 + 3i\end{align*}
    2. \begin{align*}-2 + 9i\end{align*}
    3. \begin{align*}7 - i\end{align*}
    4. \begin{align*}-5 - 2i\end{align*}
  4. Graph the complex number \begin{align*}3(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4})\end{align*} and express it in standard form.
  5. Find the standard form of each of the complex numbers below.
    1. \begin{align*}2 \ cis \frac{\pi}{2}\end{align*}
    2. \begin{align*}4 \angle \frac{5\pi}{6}\end{align*}
    3. \begin{align*}8 \left( \cos \left(-\frac{\pi}{3}\right)+i \sin \left(-\frac{\pi}{3}\right)\right)\end{align*}

Review Answers

    1. \begin{align*}5 \ cis \frac{\pi}{6}=5 \angle \frac{\pi}{6}=5 \left(\cos \frac{\pi}{6}+ i \sin \frac{\pi}{6}\right)\end{align*}
    2. \begin{align*}3 \angle 135^\circ=3cis 135^\circ=3(\cos 135^\circ+i \sin 135^\circ)\end{align*}
    3. \begin{align*}2 \left(\cos \frac{2\pi}{3}+ i \sin \frac{2\pi}{3}\right)=2cis \frac{2\pi}{3}=2 \angle \frac{2\pi}{3}\end{align*}
  1. \begin{align*}6 - 8i\end{align*} \begin{align*}& \quad 6-8i\\ x &= 6 \ \text{and} \ y=-8 && \tan \theta = \frac{y}{x}\\ r &= \sqrt{x^2+y^2} && \tan \theta = \frac{-8}{6}\\ r &= \sqrt{(6)^2+(-8)^2} && \quad \quad \theta = -53.1^\circ\\ r &=10\end{align*} Since \begin{align*}\theta\end{align*} is in the fourth quadrant then \begin{align*}\theta = -53.1^\circ + 360^\circ = 306.9^\circ\end{align*} Expressed in polar form \begin{align*}6 - 8i\end{align*} is \begin{align*}10(\cos 306.9^\circ + i \sin 306.9^\circ)\end{align*} or \begin{align*}10 \angle 306.9^\circ\end{align*}
    1. \begin{align*}4+3i \rightarrow x=4, y=3\end{align*}\begin{align*}r=\sqrt{4^2+3^2}=5, \tan \theta =\frac{3}{4} \rightarrow \theta=36.87^\circ \rightarrow 5(\cos 36.87^\circ + i \sin 36.87^\circ)\end{align*}
    2. \begin{align*}-2+9i \rightarrow x=-2, y=9\end{align*}\begin{align*}r=\sqrt{(-2)^2+9^2}=\sqrt{85} \approx 9.22, \tan \theta =-\frac{9}{2} \rightarrow \theta=102.53^\circ \rightarrow 9.22(\cos 102.53^\circ + i \sin 102.53^\circ)\end{align*}
    3. \begin{align*}7-i \rightarrow x=7, y=-1\end{align*}\begin{align*}r=\sqrt{7^2+1^2}=\sqrt{50} \approx 7.07, \tan \theta =-\frac{1}{7} \rightarrow \theta=351.87^\circ \rightarrow 7.07(\cos 351.87^\circ + i \sin 351.87^\circ)\end{align*}
    4. \begin{align*}-5-2i \rightarrow x=-5, y=-2\end{align*}\begin{align*}r=\sqrt{(-5)^2+(-2)^2}=\sqrt{29} \approx 5.39, \tan \theta =\frac{2}{5} \rightarrow \theta=201.8^\circ \rightarrow 5.39(\cos 201.8^\circ + i \sin 201.8^\circ)\end{align*}
  2. Note: The range of a graphing calculator’s
  3. \begin{align*}\tan^{-1}\end{align*}
  4. function is limited to Quadrants I and IV, and for points located in the other quadrants, such as
  5. \begin{align*}-2 + 9i\end{align*}
  6. in part b (in Quadrant II), you must add
  7. \begin{align*}180^\circ\end{align*}
  8. to get the correct angle
  9. \begin{align*}\theta\end{align*}
  10. for numbers given in polar form.
  11. \begin{align*}& 3 \left (\cos \frac{\pi}{4}+i \sin \frac{\pi}{4} \right )\\ & r=3\\ & x=\cos \frac{\pi}{4}=\frac{\sqrt{2}}{2}\\ & y=\sin \frac{\pi}{4}=\frac{\sqrt{2}}{2}\end{align*} The standard form of the polar complex number \begin{align*}3 \left (\cos \frac{\pi}{4}+i \sin \frac{\pi}{4} \right )\end{align*} is \begin{align*}\frac{3\sqrt{2}}{2}+\frac{3\sqrt{2}}{2}i\end{align*}.
    1. \begin{align*}2cis \frac{\pi}{2} \rightarrow x=\cos \frac{\pi}{2}=0, y=\sin \frac{\pi}{2}=1 \rightarrow 2(0)+2(1i)=2i\end{align*}
    2. \begin{align*}4 \angle \frac{5\pi}{6} \rightarrow x=\cos \frac{5\pi}{6}=-\frac{\sqrt{3}}{2}, y= \sin \frac{5\pi}{6}=\frac{1}{2} \rightarrow 4 \left(-\frac{\sqrt{3}}{2}\right)+4\left(i\frac{1}{2}\right)=-2 \sqrt{3}+2i\end{align*}
    3. \begin{align*}8\left(\cos \left(-\frac{\pi}{3}\right)+i \sin \left(-\frac{\pi}{3}\right)\right) \rightarrow x=\cos \left(-\frac{\pi}{3}\right)=\frac{1}{2}, y=\sin \left(-\frac{\pi}{3}\right)=-\frac{\sqrt{3}}{2} \rightarrow 8 \left(\frac{1}{2}\right)+8\left(-\frac{\sqrt{3}}{2}i\right)=4-4i\sqrt{3}\end{align*}

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