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Trigonometric Form of Complex Numbers

x plus y times i equals r times the sum of cosine theta and i times sine theta.

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Trigonometric Form of Complex Numbers

You have begun working with complex numbers in your math class. While describing numbers in the complex plane, you realize that the plotting of a complex number is a lot like plotting a set of points on a rectangular coordinate system.

You also learned in math class that you can convert coordinates from a rectangular system into a polar system. As you are considering this, you plot the complex number \begin{align*}2+3i\end{align*}. Can you somehow convert this into a type of polar plot that you've done before?

Trigonometric Form of Complex Numbers

A number in the form \begin{align*}a + bi\end{align*}, 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 problems below. 

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.

Let's look at a some problems that involve the trigonometric form of complex numbers. 

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

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*}. Finally, there is a third way to write a complex number, in the form of \begin{align*}r cis \theta\end{align*}, where "r" is the length of the vector in polar form, and \begin{align*}\theta\end{align*} is the angle the vector makes with the positive "x" axis. This makes a total of 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*}

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

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

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

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

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

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

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

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

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 #2 \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 1

Earlier, you were asked to convert a complex number into polar form. 

You can now convert \begin{align*}2+3i\end{align*} into polar form by using the equations giving the radius and angle of the number's position in the complex plane:

\begin{align*}r &= \sqrt{x^2+y^2}\\ r &= \sqrt{(2)^2+(3)^2}\\ r &= \sqrt{13}\end{align*}

\begin{align*}\tan \theta &= \frac{y}{x}\\ \tan \theta &=\frac{3}{2}\\ \theta&=56.31^\circ\end{align*}

Therefore, the polar form of \begin{align*}2+3i\end{align*} is \begin{align*}\sqrt{13}(\cos 56.31^\circ + i\sin 56.31^\circ)\end{align*}.

Example 2

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.)

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.

Example 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*}
  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*}

Example 4

Express the complex number \begin{align*}6 - 8i\end{align*} graphically and write it in its polar form.

\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*}


Plot each of the following points in the complex plane.

  1. \begin{align*}1+i\end{align*}
  2. \begin{align*}2-3i\end{align*}
  3. \begin{align*}-2-i\end{align*}
  4. \begin{align*}i\end{align*}
  5. \begin{align*}4-i\end{align*}

Find the trigonometric form of the complex numbers where \begin{align*}0\leq \theta <2\pi\end{align*}.

  1. \begin{align*}8-6i\end{align*}
  2. \begin{align*}5+12i\end{align*}
  3. \begin{align*}2-2i\end{align*}
  4. \begin{align*}3+3i\end{align*}
  5. \begin{align*}2+3i\end{align*}
  6. \begin{align*}5-6i\end{align*}

Write each complex number in standard form.

  1. \begin{align*}4(\cos 30^\circ+i\sin 30^\circ )\end{align*}
  2. \begin{align*}3(\cos \frac{\pi}{4}+i\sin \frac{\pi}{4} )\end{align*}
  3. \begin{align*}2(\cos \frac{7\pi}{6}+i\sin \frac{7\pi}{6} )\end{align*}
  4. \begin{align*}2(\cos \frac{\pi}{12}+i\sin \frac{\pi}{12} )\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.9. 

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rcis\theta rcis\theta is shorthand for the expression r\cos\theta+ri\sin\theta.
complex plane The complex plane is the graphical representation of the set of all complex numbers.
polar form The polar form of a point or a curve is given in terms of r and \theta and is graphed on the polar plane.
rectangular form The rectangular form of a point or a curve is given in terms of x and y and is graphed on the Cartesian plane.
trigonometric form To write a complex number in trigonometric form means to write it in the form r\cos\theta+ri\sin\theta. rcis\theta is shorthand for this expression.
trigonometric polar form To write a complex number in trigonometric form means to write it in the form r\cos\theta+ri\sin\theta. rcis\theta is shorthand for this expression.

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