Let $M (x,y)$ be the point in the complex plane which is joined to the complex number $z = x + yi$.
We can determine the position of the point $M$ (and thus of the complex number $z$) by using numbers $r$ and $\varphi$, where $r = |z|= \sqrt{x^2+y^2}$ (the distance from the point $M$ to the origin) and $\varphi \in \left [0, 2\pi \right \rangle $ is an angle between the segment line $\overline{OM}$ and positive part of the real axis. Number $r$ is called modulus of a complex number and angle $\varphi$ is called an argument of a complex number and it is denoted by $\varphi = arg(z)$.
Then
$$\cos \varphi = \frac{x}{r} \Rightarrow x = r \cos \varphi $$
and
$$\sin \varphi = \frac{y}{r} \Rightarrow y = r \sin \varphi$$
is valid.
Substituting in the expression $z=x + yi$, we obtain the trigonometric form of the complex number:
$$ z= r (\cos \varphi + i \sin \varphi).$$
If a complex number is given in the algebraic form $ z = x+yi$, then we determine $r$ and $\varphi$ from equations:
$$r = \sqrt{x^2 + y^2}$$
$$tg \varphi = \frac{y}{x} , x \neq 0.$$
The last equation gives two solutions for an angle $\varphi \in \left [0, 2\pi \right \rangle$. We choose the angle depending on in which quadrant number $z$ is located.
Example 1: Write in the trigonometric form complex numbers $z$ and $\overline{z}$, if $z = \frac{1}{2} – \frac{\sqrt{3}}{2}i$.
Solution:
We need to determine numbers $r$ and $\varphi$.
$$r = |z| =\left|\frac{1}{2} – \frac{\sqrt{3}}{2}i\right| $$
$$= \sqrt{\left( \frac{1}{2} \right) ^2 + \left( -\frac{\sqrt{3}}{2} \right)^2 } $$
$$= \sqrt{\frac{1}{4} + \frac{3}{4}} $$
$$= 1. $$
$$tg \varphi = \frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}} = -\sqrt{3},$$
that is $\varphi = \frac{2\pi}{3}$ or $\varphi = \frac{5\pi}{3}$. Since the number $z = \frac{1}{2} – \frac{\sqrt{3}}{2}i$ is located in the fourth quadrant, it follows that $\varphi = \frac{5\pi}{3}$.
The complex number $z= \frac{1}{2} – \frac{\sqrt{3}}{2}i$ has the trigonometric form:
$$ z = \cos \frac{5 \pi}{3} + i \sin \frac{5\pi}{3}.$$
The complex conjugate numbers have the same modulus. Therefore, for $\overline{z} = \frac{1}{2} + \frac{\sqrt{3}}{2}i$ , $r =1$. $\overline{z}$ is located in the first quadrant, so we have:
$$tg \varphi = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \Rightarrow \varphi = \frac{\pi}{3}.$$
Finally, the complex number $\overline{z} = \frac{1}{2} + \frac{\sqrt{3}}{2}$ has the following trigonometric form:
$$\overline{z} = \cos \frac{\pi}{3} + i \sin \frac{\pi}{3}.$$
Example 2: Write in the trigonometric form the complex number $z$:
$$ z = – \cos \frac{\pi}{5} + i \sin \frac{\pi}{5}.$$
Solution:
The function cosine is negative in the second and third quadrant, and sine is positive in the first and second quadrant. This means that the given complex number $z$ is located in the second quadrant.
$r = \sqrt{\left( -cos \frac{\pi}{5}\right) ^2 + \left( sin\frac{\pi}{5} \right)^2 } = \sqrt{1} = 1$
Now we have:
$$tg \varphi = \frac{\sin \frac{\pi}{5}}{-\cos \frac{\pi}{5}} = – tg \frac{\pi}{5}.$$
That is, $ \varphi = – \frac{\pi}{5}$ or $\varphi = \frac{4\pi}{5}$.
We know that the complex number $z$ is located in the second quadrant, which means that $ \varphi = \frac{4\pi}{5}$.
Now we can write the given complex number $z$ in the trigonometric form:
$$ z = \cos \frac{4 \pi}{5} + i \sin \frac{4 \pi}{3}.$$
