The goniometric form of a complex number
Kapitoly: Complex numbers, Graphical representation of complex numbers, The goniometric form of a complex number
It is not always worthwhile to have a complex number in algebraic form, so the goniometric form of a complex number is still introduced.
How can you express a point in the plane?
We know that in geometric terms, the complex number z = x + yi represents a point in the Gaussian plane. This point has coordinates [x, y]. What other way can we define the point z besides listing the coordinates [x, y]?
We can calculate the angle that the line of the point z and the origin makes with the axis x (or its positive semi-axis). This will let us know which direction the point z lies away from the origin. To find out exactly where it lies, we still need to know the distance from the origin. With these two pieces of information, we are now able to precisely define the point z in the Gaussian plane.
Goniometric shape
The following figure summarizes what we need to know in order to express a complex number in goniometric form:
As can be seen, we need to know the length of the line from the point z to the origin, which is equal to the absolute value of the number z - we can already calculate this. We also need to know the angle $\varphi$. The goniometric form of the complex number then looks like this:
$$z=|z|(\cos\varphi+i\sin\varphi)$$
How do we find the angle $\varphi$? We use the goniometric function to do this. Here we have a right triangle and we know the length of the hypotenuse, that is the absolute value of the number z. In goniometric form we have both sine and cosine, so we need to express the angle $\varphi$ using both functions. But in doing so, from the properties of goniometric functions, it is true that:
$$\begin{eqnarray} \sin\varphi&=&\frac{y}{|z|}\\ \cos\varphi&=&\frac{x}{|z|} \end{eqnarray}$$
From these formulas we can derive the angle itself $\varphi$. Deriving the angle is necessary because we write the goniometric form including the sine and cosine.
Example
Convert a complex number in algebraic form to goniometric form: $z=\sqrt{3}+i$. The first thing we do is calculate the absolute value of the number z, which is equal to:
$$|z|=\sqrt{3+1}=2$$
So now:
$$\begin{eqnarray} \sin\varphi&=&\frac{1}{2}\\ \cos\varphi&=&\frac{\sqrt{3}}{2} \end{eqnarray}$$
At this point, we can either use a calculator to calculate the arcus sine and arcus cosine, or we can use a table of basic goniometric formulas. Both values we came up with are tabulated, so we won't use the calculator for this one.
If $\sin\varphi=\frac12$, then the angle $\varphi$ is equal to either π/6 or 5π/6. If $\cos\varphi=\frac{\sqrt{3}}{2}$, then the angle $\varphi$ is equal to either π/6 or 11π/6. The intersection of these possibilities is the angle π/6, using this angle we get the values we wrote down in the previous equation.
Multiplication and division
We can, of course, multiply or divide complex numbers in goniometric form. Using the sum goniometric formulas, we can derive formulas for the product and quotient of two complex numbers. Consider two complex numbers z1 and z2:
$$\begin{eqnarray} z_1&=&|z_1|(\cos\varphi_1+i\sin\varphi_1)\\ z_2&=&|z_2|(\cos\varphi_2+i\sin\varphi_2) \end{eqnarray}$$
The formulas themselves look like this:
$$\begin{eqnarray} z_1\cdot z_2&=&|z_1|\cdot|z_2|\left[\cos(\varphi_1+\varphi_2)+i\sin(\varphi_1+\varphi_2)\right]\\ \frac{z_1}{z_2}&=&\frac{|z_1|}{|z_2|}\left[\cos(\varphi_1-\varphi_2)+i\sin(\varphi_1-\varphi_2)\right] \end{eqnarray}$$