Question - Complex Numbers Operations in Polar Form
Solution:
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:$$ w = \sqrt{3} + i $$
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com