A calculus of the absurd

15.3 The trigonometric form of a complex number

When we say \(z := x + iy\) (also known as “z is defined as \(x + iy\)”), \(x\) and \(y\) can be anything! For example, \(x\) and \(y\) could equal \(\cos (\theta )\) and \(\sin (\theta )\) respectively. By setting different values of theta, we can obtain any point on a circle with radius 1, and centre \((0, 0)\). For example, if we have \(z := \cos (\theta ) + i \sin (\theta )\), then to obtain the number \(1\), we can just set \(\theta = 0\). To be able to obtain every complex number, however, we need to introduce a second variable (which we can call \(r\)). This “scales” the circle, so that for each value of \(r\), the values of \(\theta \) between \(0\) and \(2\pi \) (\(2\pi \) is not inclusive) we can obtain a circle with that radius. Overall, we can write any complex number in the form

\begin{equation} z := r \rbrackets {\cos (\theta ) + i \sin (\theta )} \end{equation}

To find the trigonometric form of a given complex number there are two ways.

To convert a complex number (e.g. \(1 + 3i\)) into trigonometric form, the first way algebra is to use algebra (in particular “comparing coefficients”). By comparing \(r\cos (\theta ) + ri\sin (\theta )\) with \(1 + 3i\), we obtain that \(r \cos (\theta ) = 1\) and that \(r\sin (\theta ) = 3\). Thus

\begin{align} r^2 \cos ^2(\theta ) + r^2 \sin ^2(\theta ) &= r^2(\cos ^2(\theta ) + \sin ^2(\theta ) \notag \\ &= r^2 \notag \\ &= 2 \text { (from $1+3i$)} \notag \end{align}

from which we can deduce that \(r=\pm \sqrt {2}\). The other thing we can do is divide through, thus obtaining that \(\frac {r \sin (\theta )}{r \cos (\theta )} = \frac {1}{1}\), and thus that \(\theta = \arctan (1)\) (which is \(\frac {\sqrt {2}}{2}\)). Overall, we can then write that

\begin{equation} 1 + i = \sqrt {2} e ^{\frac {\sqrt {2}}{2}i} \end{equation}

which we can also do for any complex number.

The second way involves geometry (I still need to find where I originally wrote my notes on this, but the method is to draw the complex number - not necessary, but usually helpful - and to then find the modulus and argument of the complex number).