complex numbers
I need to express $z = 4e^{-i\pi/3}$ in the form of $x + yi$ and represent it on the Argand diagram.
I think that $4 = \sqrt{x^{2} + y^{2}}$ and that $\tan (\pi/3) = y/x$ but I haven't been able to do anything useful with this information…
Is this solvable via simultaneous equations?
Thanks!
The problem is that points can be expressed in polar form in more than one way. Take a look at the diagram below:
There's your point (in blue). As you can see, it's in the second quadrant, and the angle $\theta = \frac{2\pi}{3}$ passes through it. So one could accurately say that the point's polar representation is $r = 1, \theta = \frac{2\pi}{3}$, i.e. $$z = \cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right)$$
You'll notice, though, that $\theta = -\frac{\pi}{3}$ represents the same angle, but in the opposite direction. With this in mind, one could just as accurately say that the point's polar representation is $r = -1, \theta = -\frac{\pi}{3}$, i.e. $$z = -\cos\left(-\frac{\pi}{3}\right) - i\sin\left(-\frac{\pi}{3}\right)$$
Because $\cos\left(\frac{2\pi}{3}\right) = -\cos\left(-\frac{\pi}{3}\right)$ and $\sin\left(\frac{2\pi}{3}\right) = -\sin\left(-\frac{\pi}{3}\right)$, these two polar representations are equivalent. In other words, there's nothing wrong with concluding, as you did, that $\theta = -\frac{\pi}{3}$ You and the so-called "required answer" can reasonably disagree here and both be correct.
But you still did something wrong. If you choose to use $\theta = -\frac{\pi}{3}$, then you must also choose $r = -1$, yet you choose $r = 1$, based on your modulus computation. How can you avoid making this mistake in the future?
Simple: look at the quadrant in which the point lies. The point $-\frac{1}{2} + \frac{\sqrt{3}}{2}i$ has negative real part and positive imaginary part. Therefore it must be in the second quadrant. The angle $\theta = -\frac{\pi}{3}$ extends into the fourth quadrant, so if you intend to use that angle, then you must also make $r$ negative.
Hint: Rewrite as $$\left|z-(-2+2\sqrt{3}i)\right|=2.$$ This "says" that the distance from $z$ to $-2+2\sqrt{3}i$ is $2$. So the locus is the circle with a certain centre, a certain radius.
Draw that circle (crucial). You are interested in lines from the origin to points on your circle. The maximum, minimum angles are at points of tangency. One of them will be obvious from the picture. You can work out the other using once familiar geometry/trigonometry. Note that it is $\text{arg}(z)$ that you are being asked about.
Best Answer
Hint
Use the Euler's formula $$e^{i\theta}=\cos\theta+i\sin\theta$$