Trigonometry and the Complex Plane

This section covers:

In certain physics and engineering applications, it’s easier to perform certain computations with complex numbers (numbers that give a negative result when squared), rather than with real numbers (“normal” numbers).

In this section, we’ll learn how to convert Rectangular Form coordinates and equations to Polar (Trig) Complex Form, in order to perform these computations.

Review of Complex Numbers

We learned that complex numbers exist so we can do certain computations in math, even though conceptually the numbers aren’t “real”. We learned about them here in the Imaginary (Non-Real) and Complex Numbers section.

To work with complex numbers and trig, we need to learn about how they can be represented on a coordinate system (complex plane), with the “\(x\)”-axis being the real part of the point or coordinate, and the “\(y\)”-axis being the imaginary part of the point. For example, the representation of the complex number \(z=x+yi\) would be \((x,y)\) in the complex plane.

Here is a visual representation of a point in the complex plane, with it’s magnitude. The magnitude (sometimes called modulus) of a complex number is like the hypotenuse of a triangle, with lines drawn to the \(x\) (real) and \(y\) (imaginary) coordinates as the sides of the triangles (see above). Thus, the magnitude of \((4,3)\) or \(4+3i\) is \(\sqrt{{{{4}^{2}}+{{3}^{3}}}}=\sqrt{{25}}=5\).


This form of a trig number is sometimes written as \(z=a+bi\) instead of \(z=x+yi\)  (in this case, \(z=4+3i\)). And remember that \(i=\sqrt{{-1}}\), although we won’t really need to use this in this section.

You can also put complex numbers in your graphing calculator, and even perform complex conversions on the graphing calculator later.

Polar (Trig) Form of a Complex Number

Again, we can write the rectangular form of a complex number in the form \(z=x+yi\), or more commonly, \(z=a+bi\).

To write this in a trig polar form, where \(x=r\cos \theta \) and \(y=r\sin \theta \) we have the following. Notice we can use the abbreviation cis or CIS (for cos plus i sin), since the angles measurements (\(\theta \)) are the same.


Note that “\(r\)” is called the magnitude or modulus of \(z\), like we saw earlier, and is sometimes written as \(\left| z \right|\). The angle \(\theta \) is called the argument.

\(\displaystyle z=x+yi=\left( {r\cos \theta } \right)+\left( {r\sin \theta } \right)i\) which can be abbreviated to \(\displaystyle z=r\cdot \text{cis}\left( \theta \right)\) or \(\displaystyle z=r\cdot \text{cis}\theta \)

Note that \(\displaystyle \left( {r\cos \theta } \right)+\left( {r\sin \theta } \right)i\)  actually turns out to be the equivalent of \(r{{e}^{{\theta \,i}}}\) (Euler’s equation); in fact, we’ll see in the calculator section below that sometimes we need to input it this way. (We won’t prove this here, or use it any more).

Here is a visual example; note that we get the cosine and sine of 45° from the Unit Circle, where we learned about here in the Angles and the Unit Circle section:

Converting Complex Rectangular Form to Polar Form

When converting from Rectangular to Polar Form, to get \(r\) and \(\theta \), we have to use the same equations we did here in the Polar Coordinates, Equations and Graphs section here:

\(\displaystyle r=\sqrt{{{{x}^{2}}+{{y}^{2}}}}\,\,\,\,\,\,\,\,\,\text{(this will be positive)}\)

\(\displaystyle \theta ={{\tan }^{{-1}}}\left( {\frac{y}{x}} \right)\,\,\,\,\,\,\text{(check for correct quadrant)}\)

Note that when using the calculator to get \({{\tan }^{{-1}}}\), you’ll have to add the following degrees or radians when your point is in the following quadrants. This is because the \({{\tan }^{{-1}}}\) function on your calculator only give you answers back in the interval \(\displaystyle \left( {-\frac{\pi }{2},\frac{\pi }{2}} \right)\).

Here are some examples (I’ve left the argument in degrees). Remember that \(x\) is the real number and \(y\) is the imaginary number (coefficient of “\(i\)”).

Rectangular Complex Form Convert to Polar Complex Form
\(-1+\sqrt{3}i\) \(\begin{array}{c}r=\sqrt{{{{x}^{2}}+{{y}^{2}}}}=\sqrt{{{{{\left( {-1} \right)}}^{2}}+{{{\left( {\sqrt{3}} \right)}}^{2}}}}=\sqrt{{1+3}}=\sqrt{4}=2\\\theta ={{\tan }^{{-1}}}\left( {\frac{{\sqrt{3}}}{{-1}}} \right)=120{}^\circ \text{ (2nd quadrant)}\\2\left( {\cos 120{}^\circ +i\sin 120{}^\circ } \right)=2\,\text{cis}\left( {120{}^\circ } \right)\end{array}\)     
\(-1-i\) \(\begin{array}{c}r=\sqrt{{{{x}^{2}}+{{y}^{2}}}}=\sqrt{{{{{\left( {-1} \right)}}^{2}}+{{{\left( {-1} \right)}}^{2}}}}=\sqrt{{1+1}}=\sqrt{2}\\\theta ={{\tan }^{{-1}}}\left( {\frac{{-1}}{{-1}}} \right)=225{}^\circ \text{ (3rd quadrant)}\\\sqrt{2}\left( {\cos 225{}^\circ +i\sin 225{}^\circ } \right)=\sqrt{2}\,\text{cis}\left( {225{}^\circ } \right)\end{array}\)             
\(-3\) \(\displaystyle \begin{array}{c}r=\sqrt{{{{x}^{2}}+{{y}^{2}}}}=\sqrt{{{{{\left( {-3} \right)}}^{2}}+0{}^{{^{2}}}}}=\sqrt{9}=3\\\theta ={{\tan }^{{-1}}}\left( {\frac{0}{{-3}}} \right)=180\text{ (between 2nd }\\\text{ and 3rd quadrants)}\\3\left( {\cos 180{}^\circ +i\sin 180{}^\circ } \right)=3\,\text{cis}\left( {180{}^\circ } \right)\end{array}\)             
\(i\) \(\displaystyle \begin{array}{c}r=\sqrt{{{{x}^{2}}+{{y}^{2}}}}=\sqrt{{0{}^{2}+{{1}^{2}}}}=\sqrt{1}=1\\\theta ={{\tan }^{{-1}}}\left( {\frac{1}{0}} \right)=\text{undefined (between 1st }\\\text{ and 2nd quadrants: 90}{}^\circ \text{)}\\\cos 90{}^\circ +i\sin 90{}^\circ =\text{cis}\left( {90{}^\circ } \right)\end{array}\)             
\(2-5i\) \(\displaystyle \begin{array}{c}r=\sqrt{{{{x}^{2}}+{{y}^{2}}}}=\sqrt{{{{2}^{2}}+{{{\left( {-5} \right)}}^{2}}}}=\sqrt{{4+25}}=\sqrt{{29}}\\\theta ={{\tan }^{{-1}}}\left( {\frac{{-5}}{2}} \right)=291.8{}^\circ \text{ (calculator gives us -68}\text{.2}{}^\circ \text{, but we }\\\text{ need to add 360}{}^\circ \text{, since in 4th quadrant)}\\\sqrt{{29}}\left( {\cos 291.8{}^\circ +i\sin 291.8{}^\circ } \right)=\sqrt{{29}}\,\text{cis}\left( {291.8{}^\circ } \right)\end{array}\)             


Converting Polar Form to Rectangular Form

Now let’s write the polar complex numbers in rectangular form. To do this, you just get the sin and cos of the angle, and then push through (distribute) the number in front (the “\(r\)”). Again, this is similar to what we did in the Polar Coordinates, Equations and Graphs section here much easier going this way!


(These match up with the conversions above, and some are in radian mode instead of degree mode).

Polar Complex Form Convert to Rectangular Complex Form
\(2\,\text{cis}\left( {120{}^\circ } \right)\) \(\begin{align}2\,\text{cis}\left( {120{}^\circ } \right)&=2\left( {\cos 120+i\sin 120} \right)\\&=2\left( {-\frac{1}{2}+i\cdot \frac{{\sqrt{3}}}{2}} \right)=-1+\sqrt{3}i\end{align}\)
\(\displaystyle \sqrt{2}\,\text{cis}\left( {\frac{{5\pi }}{4}} \right)\) \(\begin{align}\sqrt{2}\,\text{cis}\left( {\frac{{5\pi }}{4}} \right)&=\sqrt{2}\left( {\cos \frac{{5\pi }}{4}+i\sin \frac{{5\pi }}{4}} \right)\\&=\sqrt{2}\left( {-\frac{{\sqrt{2}}}{2}+i\cdot -\frac{{\sqrt{2}}}{2}} \right)=-1-i\end{align}\)
\(3\,\text{cis}\left( \pi \right)\) \(\begin{align}3\,\text{cis}\left( \pi \right)&=3\left( {\cos \pi +i\sin \pi } \right)\\&=3\left( {-1+i\cdot 0} \right)=-3\end{align}\)
\(\text{cis}\left( {90{}^\circ } \right)\) \(\begin{align}\text{cis}\left( {90{}^\circ } \right)&=\cos 90{}^\circ +i\sin 90{}^\circ \\&=0+i\cdot 1=i\end{align}\)
\(\displaystyle \sqrt{{29}}\,\text{cis}\left( {291.8{}^\circ } \right)\) \(\begin{align}\sqrt{{29}}\,\text{cis}\left( {291.8{}^\circ } \right)&=\sqrt{{29}}\left( {\cos 291.8{}^\circ +i\sin 291.8{}^\circ } \right)\\&\approx 2-5i\end{align}\)

Products and Quotients of Complex Numbers in Polar Form

You can use the polar form of a complex number to find products and quotients of complex numbers; you’ll basically want to memorize these formulas.

Just remember that when you multiply two polar complex numbers, you multiply the magnitudes (numbers in front), but add the angle measurements. When you divide two polar complex numbers, you divide the magnitudes (numbers in front), but subtract the angle measurements. Pretty weird, huh?

\(\displaystyle {{z}_{1}}={{r}_{1}}\left( {\cos {{\theta }_{1}}+i\sin {{\theta }_{1}}} \right)=\,{{r}_{1}}\,\text{cis}\left( {{{\theta }_{1}}} \right)\,\,\,\,\,\,\,\,\text{and}\,\,\,\,\,\,\,\,{{z}_{2}}=\,\,{{r}_{2}}\left( {\cos {{\theta }_{2}}+i\sin {{\theta }_{2}}} \right)={{r}_{2}}\,\text{cis}\left( {{{\theta }_{2}}} \right)\)


\(\displaystyle \text{Product:}\,\,\,\,{{z}_{1}}{{z}_{2}}={{r}_{1}}{{r}_{2}}\left[ {\cos \left( {{{\theta }_{1}}+{{\theta }_{2}}} \right)+i\sin \left( {{{\theta }_{1}}+{{\theta }_{2}}} \right)} \right]=\,{{r}_{1}}r\,\text{cis}\left( {{{\theta }_{1}}+{{\theta }_{2}}} \right)\)

\(\displaystyle \text{Quotient:}\,\,\,\,\,\frac{{{{z}_{1}}}}{{{{z}_{2}}}}=\frac{{{{r}_{1}}}}{{{{r}_{2}}}}\left[ {\cos \left( {{{\theta }_{1}}-{{\theta }_{2}}} \right)+i\sin \left( {{{\theta }_{1}}-{{\theta }_{2}}} \right)} \right]\,=\frac{{{{r}_{1}}}}{{{{r}_{2}}}}\,\text{cis}\left( {{{\theta }_{1}}-{{\theta }_{2}}} \right)\)

Here are some examples; both types of notation are used for complex polar numbers. We’ll leave our answers in Polar Form between 0 and 360° or 0 and \(2\pi \):

Complex Numbers Products and Quotients
\(\begin{array}{c}z=2\,\text{cis}\left( {20{}^\circ } \right);\,\,\,\,\,w=4\,\text{cis}\left( {40{}^\circ } \right)\\\text{Find }zw\text{ and }\frac{z}{w}\end{array}\) \(\begin{align}zw&={{r}_{1}}{{r}_{2}}\left[ {\cos \left( {{{\theta }_{1}}+{{\theta }_{2}}} \right)+i\sin \left( {{{\theta }_{1}}+{{\theta }_{2}}} \right)} \right]\\&=2\cdot 4\cdot \text{cis}\left( {20+40} \right)=8\,\text{cis}\left( {\text{60}{}^\circ } \right)\\\frac{{{{z}_{1}}}}{{{{z}_{2}}}}&=\frac{{{{r}_{1}}}}{{{{r}_{2}}}}\left[ {\cos \left( {{{\theta }_{1}}-{{\theta }_{2}}} \right)+i\sin \left( {{{\theta }_{1}}-{{\theta }_{2}}} \right)} \right]\\&=\frac{2}{4}\cdot \text{cis}\left( {20-40} \right)=\frac{1}{2}\text{cis}\left( {-\text{20}{}^\circ } \right)=\frac{1}{2}\text{cis}\left( {34\text{0}{}^\circ } \right)\end{align}\)
\(\begin{array}{c}z=4\text{cis}\left( {\frac{\pi }{8}} \right);\,\,\,\,\,w=3\text{cis}\left( {\frac{{9\pi }}{{16}}} \right)\\\text{Find }zw\text{ and }\frac{z}{w}\end{array}\) \(\begin{align}zw&={{r}_{1}}{{r}_{2}}\left[ {\cos \left( {{{\theta }_{1}}+{{\theta }_{2}}} \right)+i\sin \left( {{{\theta }_{1}}+{{\theta }_{2}}} \right)} \right]\\&=4\cdot 3\cdot \text{cis}\left( {\frac{\pi }{8}+\frac{{9\pi }}{{16}}} \right)=12\,\text{cis}\left( {\frac{{11\pi }}{{16}}} \right)\\\frac{{{{z}_{1}}}}{{{{z}_{2}}}}&=\frac{{{{r}_{1}}}}{{{{r}_{2}}}}\left[ {\cos \left( {{{\theta }_{1}}-{{\theta }_{2}}} \right)+i\sin \left( {{{\theta }_{1}}-{{\theta }_{2}}} \right)} \right]\\&=\frac{4}{3}\cdot \text{cis}\left( {\frac{\pi }{8}-\frac{{9\pi }}{{16}}} \right)=\frac{4}{3}\text{cis}\left( {\frac{{-7\pi }}{{16}}} \right)=\frac{4}{3}\text{cis}\left( {\frac{{25\pi }}{{16}}} \right)\end{align}\)
\(\begin{array}{c}z=1-i;\,\,\,\,\,w=1+\sqrt{3}i\\\text{Find }zw\text{ and }\frac{z}{w}\end{array}\)



\(\begin{array}{c}z=1-i:\\\\r=\sqrt{{{{x}^{2}}+{{y}^{2}}}}=\sqrt{{{{1}^{2}}+{{{\left( {-1} \right)}}^{2}}}}=\sqrt{2}\\\theta ={{\tan }^{{-1}}}\left( {\frac{{-1}}{1}} \right)=315{}^\circ \\\text{(4th quadrant)}\\\sqrt{2}\,\text{cis}\left( {315{}^\circ } \right)\end{array}\)

\(\begin{array}{c}z=1+\sqrt{3}i:\\\\r=\sqrt{{{{x}^{2}}+{{y}^{2}}}}=\sqrt{{{{1}^{2}}+{{{\left( {\sqrt{3}} \right)}}^{2}}}}=2\\\theta ={{\tan }^{{-1}}}\left( {\frac{{\sqrt{3}}}{1}} \right)=60{}^\circ \\\text{ (1st quadrant)}\\2\,\text{cis}\left( {60{}^\circ } \right)\end{array}\)
\(\displaystyle \begin{align}zw&={{r}_{1}}{{r}_{2}}\left[ {\cos \left( {{{\theta }_{1}}+{{\theta }_{2}}} \right)+i\sin \left( {{{\theta }_{1}}+{{\theta }_{2}}} \right)} \right]\\&=\sqrt{2}\cdot 2\,\text{cis}\left( {315+60} \right)=2\sqrt{2}\,\text{cis}\left( {375{}^\circ } \right)=2\sqrt{2}\,\text{cis}\left( {15{}^\circ } \right)\\\frac{{{{z}_{1}}}}{{{{z}_{2}}}}&=\frac{{{{r}_{1}}}}{{{{r}_{2}}}}\left[ {\cos \left( {{{\theta }_{1}}-{{\theta }_{2}}} \right)+i\sin \left( {{{\theta }_{1}}-{{\theta }_{2}}} \right)} \right]\\&=\frac{{\sqrt{2}}}{2}\cdot \text{cis}\left( {315-60} \right)=\frac{{\sqrt{2}}}{2}\text{cis}\left( {255{}^\circ } \right)\end{align}\)

De Moivre’s Theorem: Powers of Complex Numbers

De Moivre’s Theorem (named after the French mathematician Abraham De Moivre), is a formula for raising a complex number to a power (greater than or equal to 1).

Just remember that when you raise a complex number to a power, you raise the magnitude to that power, but multiply the angle measurements. (Later we’ll see that when you take the root of a complex number, you take the root of the magnitude, but divide the angle measurements).

See how it’s similar to multiplying and dividing complex numbers?

Here is De Moivre’s Theorem:

\(\displaystyle {{z}^{n}}={{r}^{n}}\left[ {\cos \left( {n\theta } \right)+i\sin \left( {n\theta } \right)} \right]\,={{r}^{n}}\left[ {\text{cis}\left( {n\theta } \right)} \right]\)

Here are some examples; both types of notation are used for complex polar numbers. We’ll write our answers in Standard or Rectangular Form \(a+bi\):

Complex Numbers Finding Powers using De Moivre’s Theorem
\({{\left[ {2\,\text{cis}\left( {80{}^\circ } \right)} \right]}^{{3}}}\) \(\displaystyle {{\left[ {2\,\text{cis}\left( {80{}^\circ } \right)} \right]}^{{\,3}}}={{2}^{3}}\,\text{cis}\left( {3\cdot 80{}^\circ } \right)=8\,\text{cis}\left( {\text{240}{}^\circ } \right)=8\left( {-\frac{1}{2}-\frac{{\sqrt{3}}}{2}i} \right)=-4-4\sqrt{3}i\)
\(\displaystyle {{\left[ {\sqrt{3}\,\text{cos}\left( {\frac{{7\pi }}{{18}}} \right)+i\sin \left( {\frac{{7\pi }}{{18}}} \right)} \right]}^{{6}}}\) \(\displaystyle \begin{align}{{\left[ {\sqrt{3}\,\text{cos}\left( {\frac{{7\pi }}{{18}}} \right)+i\sin \left( {\frac{{7\pi }}{{18}}} \right)} \right]}^{{\,6}}}&={{\left( {\sqrt{3}} \right)}^{6}}\text{cis}\left( {6\cdot \frac{{7\pi }}{{18}}} \right)={{3}^{{\frac{6}{2}}}}\text{cis}\left( {\frac{{42\pi }}{{18}}} \right)\\&=27\,\text{cis}\left( {\frac{{42\pi }}{{18}}-\frac{{36\pi }}{{18}}} \right)=27\,\text{cis}\left( {\frac{\pi }{3}} \right)\\&=27\left( {\frac{1}{2}+\frac{{\sqrt{3}}}{2}i} \right)=\frac{{27}}{2}+\frac{{27\sqrt{3}}}{2}i\end{align}\)
\({{\left( {1-i} \right)}^{7}}\)         Convert to Polar first:              \(\begin{array}{c}z=1-i:\\r=\sqrt{{{{x}^{2}}+{{y}^{2}}}}=\sqrt{{{{1}^{2}}+{{{\left( {-1} \right)}}^{2}}}}=\sqrt{2}\\\theta ={{\tan }^{{-1}}}\left( {\frac{{-1}}{1}} \right)=315{}^\circ \text{(4th quadrant)}\\\sqrt{2}\text{cis}\left( {315{}^\circ } \right)\end{array}\)
\(\displaystyle \begin{align}{{\left[ {\sqrt{2}\text{cis}\left( {315{}^\circ } \right)} \right]}^{{\,7}}}&={{\left( {\sqrt{2}} \right)}^{7}}\text{cis}\left( {7\cdot 315{}^\circ } \right)={{2}^{{\frac{7}{2}}}}\text{cis}\left( {2205{}^\circ } \right)={{2}^{3}}\cdot {{2}^{{\frac{1}{2}}}}\text{cis}\left( {2205{}^\circ -360{}^\circ \cdot 6} \right)\\&=8\sqrt{2}\,\text{cis}\left( {45{}^\circ } \right)=8\sqrt{2}\left( {\frac{{\sqrt{2}}}{2}+\frac{{\sqrt{2}}}{2}i} \right)=8+8i\end{align}\)

Roots of Complex Numbers

To get roots of complex numbers, we do the opposite of raising them to a power; we take the \(n\)th root of the magnitude, and then divide the angle measurements by \(n\).

The only thing that’s a little tricky is there are typically many roots for a complex number, so we have to find all of these by the following formula, with \(k\) going from 0 to \((n-1)\):

\(\displaystyle \begin{align}\sqrt[n]{z}&=\sqrt[n]{r}\left[ {\cos \left( {\frac{\theta }{n}+\frac{{2\pi k}}{n}} \right)+i\sin \left( {\frac{\theta }{n}+\frac{{2\pi k}}{n}} \right)} \right],\,\,\,\,k=0,\,\,1,\,\,2,\,…,n-1\\&=\sqrt[n]{r}\,\text{cis}\left( {\frac{\theta }{n}+\frac{{2\pi k}}{n}} \right),\,\,\,\,k=0,\,\,1,\,\,2,\,…,n-1\end{align}\)

This looks really complicated, so let’s go through some examples (and get answers in either degrees or radians, per indicated). Let’s first work with roots with either the real or imaginary part of the complex number (but not both). For those roots that end up with special values (values on the Unit Circle), put the answers in \(a+bi\) form; for non-special values, leave in polar form.


You might also note that the roots of complex numbers are evenly spaced on the unit circle, all with the same magnitude.

Note that if a problem asks for a root of “unity”, this is just 1 \((1+0i)\).

Complex Numbers Finding Complex Roots



  Convert  \(z=-27\)  to Polar first:

\(\displaystyle r=\sqrt{{{{{\left( {-27} \right)}}^{2}}+{{0}^{2}}}}=27;\,\,\,\,\,\,\,\theta ={{\tan }^{{-1}}}\left( {\frac{0}{{-27}}} \right)=\pi :\,\,\,\, 27\text{cis}\left( \pi \right)\)       

\(\displaystyle \begin{array}{c}{{\left[ {27\text{cis}\left( \pi \right)} \right]}^{{\,\frac{1}{3}}}}={{27}^{{\frac{1}{3}}}}\text{cis}\left( {\frac{{\pi +2\pi k}}{3}} \right)\\=3\,\text{cis}\left( {\frac{{\pi +2\pi k}}{3}} \right);\,\,\,k=0,1,2\end{array}\)            \(\displaystyle \begin{array}{l}k=0:\,\,\,\,3\,\text{cis}\left( {\frac{{\pi +0}}{3}} \right)=3\,\text{cis}\left( {\frac{\pi }{3}} \right)=3\left( {\frac{1}{2}+\frac{{\sqrt{3}}}{2}i} \right)=\frac{3}{2}+\frac{{3\sqrt{3}}}{2}i\\k=1:\,\,\,\,3\,\text{cis}\left( {\frac{{\pi +2\pi }}{3}} \right)=3\,\text{cis}\left( \pi \right)=3\left( {-1+0i} \right)=-3\\k=2:\,\,\,\,3\,\text{cis}\left( {\frac{{\pi +4\pi }}{3}} \right)=3\,\text{cis}\left( {\frac{{5\pi }}{3}} \right)=3\left( {\frac{1}{2}-\frac{{\sqrt{3}}}{2}i} \right)=\frac{3}{2}-\frac{{3\sqrt{3}}}{2}i\end{array}\)



Convert  \(z=64i\)  to Polar first:

\(\displaystyle r=\sqrt{{{{0}^{2}}+{{{\left( {-64} \right)}}^{2}}+}}=64;\,\,\,\theta ={{\tan }^{{-1}}}\left( {\frac{{-64}}{0}} \right)\,\,\,\text{(und)}=\text{270}{}^\circ :\,\, 64\,\text{cis}\left( {\text{270}{}^\circ } \right)\)       

\(\begin{array}{c}{{\left[ {64\text{cis}\left( {\text{270}{}^\circ } \right)} \right]}^{{\frac{1}{3}}}}={{64}^{{\frac{1}{3}}}}\text{cis}\left( {\frac{{270+360k}}{3}} \right)\\=4\text{cis}\left( {\frac{{270+360k}}{3}} \right);\,\,\,\,\,k=0,1,2\end{array}\)            \(\begin{array}{l}k=0:\,\,\,\,4\,\text{cis}\left( {90{}^\circ } \right)=4\left( {0+i} \right)=\,\,4i\\k=1:\,\,\,\,4\,\text{cis}\left( {210{}^\circ } \right)=4\left( {-\frac{{\sqrt{3}}}{2}-\frac{1}{2}i} \right)=-2\sqrt{3}-2i\\k=2:\,\,\,\,4\,\text{cis}\left( {330{}^\circ } \right)=4\left( {\frac{{\sqrt{3}}}{2}-\frac{1}{2}i} \right)=2\sqrt{3}-2i\end{array}\)
Fifth Root of Unity




Convert  \(z=1\)  to Polar first:

\(\displaystyle r=\sqrt{{{{1}^{2}}+{{0}^{2}}}}=1;\,\,\,\,\,\,\,\theta ={{\tan }^{{-1}}}\left( {\frac{0}{1}} \right)=1:\,\,\,\, \text{cis}\left( 0 \right)\)       

\(\displaystyle \begin{array}{c}{{\left[ {1\text{cis}\left( 0 \right)} \right]}^{{\,\frac{1}{5}}}}={{1}^{{\frac{1}{5}}}}\text{cis}\left( {\frac{{0+360k}}{5}} \right)\\=1\text{cis}\left( {\frac{{360k}}{5}} \right);\,\,\,\,\,k=0,1,2,3,4\end{array}\)          \(\begin{array}{l}k=0:\,\,\,\,\text{cis}\left( {\frac{{360\left( 0 \right)}}{5}} \right)=\text{cis}\left( {0{}^\circ } \right)\\k=1:\,\,\,\,\text{cis}\left( {\frac{{360\left( 1 \right)}}{5}} \right)=\text{cis}\left( {72{}^\circ } \right)\\k=2:\,\,\,\,\text{cis}\left( {\frac{{360\left( 2 \right)}}{5}} \right)=\text{cis}\left( {144{}^\circ } \right)\end{array}\)      \(\begin{array}{l}k=3:\,\,\,\,\text{cis}\left( {\frac{{360\left( 3 \right)}}{5}} \right)=\text{cis}\left( {216{}^\circ } \right)\\k=4:\,\,\,\,\text{cis}\left( {\frac{{360\left( 4 \right)}}{5}} \right)=\text{cis}\left( {288{}^\circ } \right)\end{array}\)


Here are some more examples; we’ll keep all answers in trig form (radians). Notice how we want to find common denominators when adding the fractions with \(2\pi \) in them. Also note with the second problem how we turned \(\sqrt{{32}}\) into \(\sqrt{{{{2}^{5}}}}\) into \({{\left( {\sqrt{2}} \right)}^{5}}\) so we could simplify when we take the fifth root.

Complex Numbers Finding Complex Roots

Find fourth roots of




  Convert \(\sqrt{3}+i\) to Polar first:

\(\displaystyle r=\sqrt{{{{{\left( {\sqrt{3}} \right)}}^{2}}+{{1}^{2}}}}=\sqrt{4}=2;\,\,\,\,\,\,\theta ={{\tan }^{{-1}}}\left( {\frac{1}{{\sqrt{3}}}} \right)=\frac{\pi }{6}:\,\,\,\, 2\,\text{cis}\left( {\frac{\pi }{6}} \right)\)         

\(\displaystyle \begin{array}{c}{{\left[ {2\,\text{cis}\left( {\frac{\pi }{6}} \right)} \right]}^{{\,\frac{1}{4}}}}={{2}^{{\frac{1}{4}}}}\text{cis}\left( {\frac{{\frac{\pi }{6}+2\pi k}}{4}} \right)\\=\sqrt[4]{2}\,\text{cis}\left( {\frac{\pi }{{24}}+\frac{{2\pi k}}{4}} \right);\,\,\,\,k=0,1,2,3\\=\sqrt[4]{2}\,\text{cis}\left( {\frac{\pi }{{24}}+\frac{{12\pi k}}{{24}}} \right);\,\,\,\,k=0,1,2,3\end{array}\)              \(\displaystyle \begin{array}{l}k=0:\,\,\,\,\sqrt[4]{2}\,\text{cis}\left( {\frac{\pi }{{24}}+\frac{{12\pi \cdot 0}}{{24}}} \right)=\sqrt[4]{2}\,\text{cis}\left( {\frac{\pi }{{24}}} \right)\\k=1:\,\,\,\,\sqrt[4]{2}\,\text{cis}\left( {\frac{\pi }{{24}}+\frac{{12\pi \cdot 1}}{{24}}} \right)=\sqrt[4]{2}\,\text{cis}\left( {\frac{{13\pi }}{{24}}} \right)\\k=2:\,\,\,\,\sqrt[4]{2}\,\text{cis}\left( {\frac{\pi }{{24}}+\frac{{12\pi \cdot 2}}{{24}}} \right)=\sqrt[4]{2}\,\text{cis}\left( {\frac{{25\pi }}{{24}}} \right)\\k=3:\,\,\,\,\sqrt[4]{2}\,\text{cis}\left( {\frac{\pi }{{24}}+\frac{{12\pi \cdot 3}}{{24}}} \right)=\sqrt[4]{2}\,\text{cis}\left( {\frac{{37\pi }}{{24}}} \right)\end{array}\)





(This is the same as taking the fifth roots of \(4-4i\))


Convert \(4-4i\) to Polar:

\(\displaystyle r=\sqrt{{{{4}^{2}}+{{{\left( {-4} \right)}}^{2}}}}=\sqrt{{32}}={{\left( {\sqrt{2}} \right)}^{5}};\,\,\,\theta ={{\tan }^{{-1}}}\left( {\frac{{-4}}{4}} \right)=\frac{{7\pi }}{4}:\,\, {{\left( {\sqrt{2}} \right)}^{5}}\text{cis}\left( {\frac{{7\pi }}{4}} \right)\)       

\(\require {cancel} \displaystyle \begin{array}{c}{{\left[ {{{{\left( {\sqrt{2}} \right)}}^{5}}\text{cis}\left( {\frac{{7\pi }}{4}} \right)} \right]}^{{\,\frac{1}{5}}}}={{\left( {{{{\left( {\sqrt{2}} \right)}}^{{\cancel{5}}}}} \right)}^{{\frac{1}{{\cancel{5}}}}}}\text{cis}\left( {\frac{{\frac{{7\pi }}{4}+2\pi k}}{5}} \right)\\\,\,=\sqrt{2}\text{cis}\,\left( {\frac{{7\pi }}{{20}}+\frac{{2\pi k}}{5}} \right);\,\,\,\,k=0,1,2,3,4\\=\sqrt{2}\text{cis}\,\left( {\frac{{7\pi }}{{20}}+\frac{{8\pi k}}{{20}}} \right);\,\,\,\,k=0,1,2,3,4\end{array}\)            \(\begin{array}{l}k=0:\,\,\,\,\sqrt{2}\,\text{cis}\left( {\frac{{7\pi }}{{20}}+\frac{{8\pi \cdot 0}}{{20}}} \right)=\sqrt{2}\,\text{cis}\left( {\frac{{7\pi }}{{20}}} \right)\\k=1:\,\,\,\,\sqrt{2}\,\text{cis}\left( {\frac{{7\pi }}{{20}}+\frac{{8\pi \cdot 1}}{{20}}} \right)=\sqrt{2}\,\text{cis}\left( {\frac{{3\pi }}{{4}}} \right)\\k=2:\,\,\,\,\sqrt{2}\,\text{cis}\left( {\frac{{7\pi }}{{20}}+\frac{{8\pi \cdot 2}}{{20}}} \right)=\sqrt{2}\,\text{cis}\left( {\frac{{23\pi }}{{20}}} \right)\\k=3:\,\,\,\,\sqrt{2}\,\text{cis}\left( {\frac{{7\pi }}{{20}}+\frac{{8\pi \cdot 3}}{{20}}} \right)=\sqrt{2}\,\text{cis}\left( {\frac{{31\pi }}{{20}}} \right)\\k=4:\,\,\,\,\sqrt{2}\,\text{cis}\left( {\frac{{7\pi }}{{20}}+\frac{{8\pi \cdot 4}}{{20}}} \right)=\sqrt{2}\,\text{cis}\left( {\frac{{39\pi }}{{20}}} \right)\end{array}\)


Here’s another type of problem you might see. Note we could have solved the second part of this problem more easily with the Complex Conjugate Root Theorem, or Conjugate Zeros Theorem that we learned about here in the Imaginary (Complex) Numbers Section.

Other Complex Root Problem


Complex Trig in the Graphing Calculator

You can use the graphing calculator to convert back and forth between polar to rectangular coordinates (like we saw in the Polar Coordinates, Equations and Graphs section), and even find complex powers and roots.

For polar-rectangular conversions, use “2nd APPS (ANGLE)”, but you won’t get the answers with the roots in them (you’ll get decimals that aren’t “exact”). Note that you have to solve for the \(x\) and \(y\), or \(r\) and \(\theta \) separately.

Make sure you have your calculator either in DEGREES or RADIANS (in MODE), depending on what you’re inputting.

We’ll work with the conversion, \(2\,\text{cis}\left( {120{}^\circ } \right)\).

Graphing Calculator Complex

You can also use the MATH CMPLX →Rect and MATH CMPLX →Polar features to convert back and forth. To go from Polar to Rectangular, you have to put it in the form \(r{{e}^{{\,\theta \,i}}}\)  and put in \(\theta \) in radians, even if the calculator is in degree mode. You also have to include the “\(i\)”:

Graphing Calculator Complex 2

To multiply and divide complex numbers, you can put these in the calculator using re^(θi) mode; let’s do this problem on the calculator: \(z=1-i;\,\,\,\,\,w=1+\sqrt{3}i;\,\,\,\,\,\text{Find }zw\text{ and }\frac{z}{w}\).

Note that we have to add 360° to the negative angle.

Note also that we can convert the product or quotient back to Rectangular by using MATH CMPLX →Rect.

Product Quotient Calculator

To raise complex numbers to powers (De Moivre’s Theorem) using the calculator, again you can put these in the calculator using re^(θi) mode; let’s do these problems on the calculator: Find \({{\left[ {2\,\text{cis}\left( {80{}^\circ } \right)} \right]}^{{\,3}}}\) and find \({{\left( {1-i} \right)}^{7}}\).

Remember, on the first one, you have to use radians to input the 80° into the re^(θi) format; multiply by \(\pi \) and divide by 180 (and don’t forget the “\(i\)” in the exponent). For that problem, we get \(-4-4\sqrt{3}i\), like we got above. Note also that we can convert the complex numbers back to Rectangular by using MATH CMPLX →Rect.

Calculator De Moivre

When getting roots of complex numbers in the calculator, you’ll only get one root, so this won’t be too helpful. But you can check all the roots you get by putting them in the calculator and taking the reverse power.

For example, if we try to get the roots for \(\sqrt[3]{{-27}}\), we only get –3. But to check our answers that we got above, we can cube them to see if we get –27:

Roots in Calculator

To check to see if 216° and 288° are fifth roots of unity (1), as other examples, we can convert them to radians and take the 5th root:

Roots in Calculator 2

(Note that the coefficient of \(i\) is a very tiny number and should be in fact “0”.)

Understand these problems, and practice, practice, practice!

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software.  You can even get math worksheets.

You can also go to the Mathway site here, where you can register, or just use the software for free without the detailed solutions.  There is even a Mathway App for your mobile device.  Enjoy!

On to Introduction to Calculus and Study Guides  – you’re ready!