Complex Numbers in Polar Form
To write a complex number in polar form, we need to calculate its magnitude (r) and the angle (theta, θ) it makes with the positive real axis.
The polar form of a complex number is given by: \( r(\cos(\theta) + i\sin(\theta)) \)
For \( w = 2 - 2i \):
a) To find the magnitude (r), use the formula \( r = \sqrt{a^2 + b^2} \), where a is the real part and b is the imaginary part. So:
\[ r_w = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \]
The angle (θ) is given by \( \theta = \arctan\left(\frac{b}{a}\right) \), however, since the complex number is in the fourth quadrant, we need to add \( 2\pi \) to the result to get the positive angle.
\[ \theta_w = \arctan\left(\frac{-2}{2}\right) = \arctan(-1) \]
In radians, \( \arctan(-1) \) corresponds to \( -\frac{\pi}{4} \), but we want the positive angle, so we add \( 2\pi \):
\[ \theta_w = -\frac{\pi}{4} + 2\pi = \frac{7\pi}{4} \]
Polar form of \( w \) would then be:
\[ w = \sqrt{8}\left(\cos\left(\frac{7\pi}{4}\right) + i\sin\left(\frac{7\pi}{4}\right)\right) \]
For \( z = \frac{\sqrt{3}}{2} + \frac{1}{2}i \):
b) The magnitude (r) is:
\[ r_z = \sqrt{\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2} = \sqrt{\frac{3}{4} + \frac{1}{4}} = \sqrt{1} = 1 \]
The angle (θ) for \( z \) is in the first quadrant, and the real and imaginary parts correspond to the sine and cosine of \( \frac{\pi}{6} \), so:
\[ \theta_z = \frac{\pi}{6} \]
Polar form of \( z \) would then be:
\[ z = 1\left(\cos\left(\frac{\pi}{6}\right) + i\sin\left(\frac{\pi}{6}\right)\right) \]
For \( z^3 \), since \( |z| = 1 \) and \( \theta = \frac{\pi}{6} \):
c) Raising \( z \) to the third power multiplies its angle by 3, and its magnitude to the power of 3. Since the magnitude of \( z \) is 1, it stays 1. Thus:
\[ z^3 = 1^3\left(\cos\left(3\cdot\frac{\pi}{6}\right) + i\sin\left(3\cdot\frac{\pi}{6}\right)\right) \]
\[ z^3 = 1\left(\cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right)\right) \]
Note that \( \cos\left(\frac{\pi}{2}\right) \) is 0, and \( \sin\left(\frac{\pi}{2}\right) \) is 1, so \( z^3 \) can be further simplified to:
\[ z^3 = i \]
For part d), we are not asked for the exponential form, but in case you need help in the future:
d) The exponential form of a complex number is given by \( r \, e^{i\theta} \). Using Euler's formula \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \), we get:
\[ w = \sqrt{8} \, e^{i\frac{7\pi}{4}} \]
