Complex Numbers Operations in Polar Form
The image shows a question involving two complex numbers in polar form along with four sub-questions (a to d) asking for the product, quotient, exponential form, and rectangular form of the complex numbers. The complex numbers in question are:
\( z = \sqrt{2}(\cos(45^\circ) + i\sin(45^\circ)) \)
\( w = 2(\cos(30^\circ) + i\sin(30^\circ)) \)
Let's go through each part of the question:
a) Write the product \( zw \) in the polar form.
The product of two complex numbers in polar form is found by multiplying their magnitudes (r) and adding their angles (θ). For the given complex numbers:
\( r_z = \sqrt{2}, \theta_z = 45^\circ \)
\( r_w = 2, \theta_w = 30^\circ \)
So the product \( zw \) has a magnitude \( r_{zw} = r_z * r_w \) and an angle \( \theta_{zw} = \theta_z + \theta_w \):
\( r_{zw} = \sqrt{2} * 2 = 2\sqrt{2} \)
\( \theta_{zw} = 45^\circ + 30^\circ = 75^\circ \)
Thus, in polar form, the product is:
\( zw = 2\sqrt{2}(\cos(75^\circ) + i\sin(75^\circ)) \)
b) Write the quotient \( \frac{z}{w} \) in the polar form.
The quotient of two complex numbers in polar form is found by dividing their magnitudes and subtracting their angles:
\( r_{\frac{z}{w}} = \frac{r_z}{r_w} \)
\( \theta_{\frac{z}{w}} = \theta_z - \theta_w \)
Therefore:
\( r_{\frac{z}{w}} = \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \)
\( \theta_{\frac{z}{w}} = 45^\circ - 30^\circ = 15^\circ \)
In polar form, the quotient is:
\( \frac{z}{w} = \frac{\sqrt{2}}{2}(\cos(15^\circ) + i\sin(15^\circ)) \)
c) Write \( z \) in the exponential form.
To express the complex number \( z \) in exponential form, use Euler's formula: \( re^{i\theta} = r(\cos(\theta) + i\sin(\theta)) \):
\( z = \sqrt{2}e^{i45^\circ} \)
Note that normally the angle would be in radians, but here we'll leave it in degrees per the format of the question.
d) Write \( w \) in the rectangular form.
The rectangular (or Cartesian) form of a complex number is \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. Using the given polar form, we calculate:
\( a_w = r_w\cos(\theta_w) = 2\cos(30^\circ) = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3} \)
\( b_w = r_w\sin(\theta_w) = 2\sin(30^\circ) = 2 \times \frac{1}{2} = 1 \)
Thus, the rectangular form of \( w \) is:
\( w = \sqrt{3} + i \)
