Write $w = \sqrt{3} – i$ in polar form.
How is $\theta = \frac{-1}{\sqrt3} \textrm{ converted into } \frac{-\pi}{6}$? I understand that $w$ lies in the fourth quadrant of the unit circle, but that's about all I understand. I tried to convert degrees to radians, but I don't think that was the smart decision (I failed).
Best Answer
You should always draw the picture when you try to find the modulus and argument.
In an Argand Diagram, the complex number $x+\mathrm i y$ corresponds to the point $(x,y)$ on the $xy$-plane.
Put the point $(\sqrt 3,-1)$ on the $xy$-plane, and join it to the origin by a chord (part of a line).
You can see the chord joining the origin to the complex number as the hypotenuse of a right-angled triangle. The length of that chord is the modulus $|z|$, which can be found using Pythagoras:
$$|z|^2 = \left(\sqrt 3\right)^2 + 1^2 = 4$$
This gives $|z| = 2$, since $|z| \ge 0$.
The argument is the angle of the chord measures from "3 o'clock", i.e. the positive real axis. Anti-clockwise rotations are measured as positive, while clockwise rotations are measure as negative.
This may seem odd, but right is positive and left is negative on the number line.
We can use basic SohCahToh Trigonometry to find the argument. First, find the angle between the chord and the $y$-axis, then subtract from $90^{\circ}$ or $\frac{1}{2}\pi$ rads to get the angle between the chord and the positive $x$-axis.
$$\tan \theta = \frac{\mbox{opp}}{\mbox{adj}} = \frac{\sqrt 3}{1} = \sqrt 3$$
The angle between the chord and the $y$-axis is then $\arctan \sqrt 3 = 60^{\circ}$ or $\frac{1}{3}\pi$ rads. That means the argument, i.e. the angle between the chord and the positive $x$-axis (remembering that clockwise is negative) is $-\frac{1}{6}\pi$ or $-30^{\circ}$.
Finally, $|z|=2$ and $\arg z = -\frac{1}{6}\pi$, so $$z = r(\cos \theta + \mathrm i \sin \theta) = 2\left(\cos\left(-\frac{1}{6}\pi\right)+\mathrm i \sin\left(-\frac{1}{6}\pi\right)\right)$$