Complex number problem- separating into real and imaginary parts!

Question

Please help with a question that I am working on just now...:)

If $z=2e^{i\theta}$ where $0<\theta<\pi$, how can I find the real and imaginary parts of $w=(z-2)/(z+2)$? Hence, how can I calculate w and deduce that this complex number always lies parallel to the real axis in the Argand plane?

The following information can be used:

$\cos(2\theta)=2\cos^2(\theta)-1$ and

$\sin(2\theta)=2\cos(\theta)\sin(\theta)$

Thanks for all your help!!

Answer 1

you can argue this geometrically as follows: i will work with $0 \le \theta \le \pi$ for $\theta$ in the lower half plane replace $\theta$ by $-\theta$

$z = 2e^{i\theta}$ lies on the circle with diameter $-2$ and $2$. that is the center of the circle is $0$ and the radius is $2$ now observe that the angle subtended by the diameter on the circle is $\pi/2$ this translates as $z - 2$ and $z + 2$ are orthogonal. we can say more $$z - 2 = 2 \sin(\theta/2)e^{i(\pi+\theta)/2}, \qquad z + 2 = 2\cos (\theta/2)e^{i\theta/2},$$ hence $$ w = \dfrac{z-2}{z+2} = i\tan (\theta/2) $$ so $w$ is a pure imaginary number therefore is parallel to imaginary axis.

Answer 2

The simplest way consists in replacing $z$ with its value, then multiplying numerator and denominator by $\mathrm e^{-\mathrm i \frac\theta2}$ and using Euler's formulae.

Answer 3

it is perhaps worthwhile to observe in passing that there is an elementary geometric interpretation of these facts.

the points $P=2, M=-2$ and $Z=2e^{i\theta}$ all lie on the circle with radius $2$ centered at $O$.

the geometrical result that the angle subtended by a diameter ($MP$) at a point on the circumference ($Z$) is a right angle shows that the lines $ZP$ and $ZM$ are orthogonal, hence the ratio of $2z-2$ to $2z+2 = 2z-(-2)$ is a purely imaginary complex number. also its magnitude is $\frac{|ZP|}{|ZM|} = \tan Z \hat MP=\tan \frac{\theta}2$ by another theorem of geometry, that the angle subtended by a chord at the circumference is a half of the angle subtended by the same chord at the center (if both angles are on the same side of the chord)