Find the modulus and argument of the following complex numbers an… - Vidyadip

Question

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form: $\frac{-16}{1+\text{i}\sqrt{3}}$

✓

Answer

The polar form of a complex number $\text{z}=\text{x}+\text{iy},$ is give by $\text{z}=|\text{z}|\big(\cos\theta+\text{i}\sin\theta\big)$ Where, $|\text{z}|=\sqrt{\text{x}^2+\text{y}^2}$ and $\text{arg(z)}=\theta=\tan^{-1}\Big(\frac{\text{b}}{\text{a}}\Big)$ Let $\text{z}=\frac{-16}{1+\text{i}\sqrt{3}}$ $=\frac{-16}{1+\text{i}\sqrt{3}}\times\frac{1-\text{i}\sqrt{3}}{1-\text{i}\sqrt{3}}$ $=\frac{-16(1-\text{i}\sqrt{3})}{(1)^2+(\sqrt{3})^2}$ $=\frac{-16(1-\text{i}\sqrt{3})}{1+3}$ $=\frac{-16}{4}(1-\text{i}\sqrt{3})$ $=-4(1-\text{i}\sqrt{3})$ $=-4+4\sqrt{3}\text{i}$ $\therefore \ |\text{z}|=\sqrt{(-4)^2+(4\sqrt{3})^2}$ $=\sqrt{16+48}$ $=\sqrt{64}$ $=8$ Here $\text{x}=-4<0$ & $\text{y}=4\text{x}3>0, \ \therefore\theta$ lies in quadrantII $=\theta=\text{arg(z)}=\tan^{-1}\Big(\frac{4\sqrt{3}}{-4}\Big)$ $=\tan^{-1}(\sqrt{3})$ $=\tan^{-1}\Big(\tan\big(\pi-\frac{\pi}{3}\big)\Big) \ \big(\therefore\tan(\pi-\theta)=-\tan\theta\big)$ $=\pi-\frac{\pi}{3}$ $=\frac{2\pi}{3}$ The polar form is given by $\text{z}=8\Big(\cos\frac{2\pi}{3}+\text{i}\sin\frac{2\pi}{3}\Big)$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "(Each question 4 marks)" from Complex Numbers→Complex Numbers — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the sum of the following series to n terms: $1^3 + 3^3 + 5^3 + 7^3 + ......$

Find the general solutions of the following equations: $\sin3\text{x}+\cos2\text{x}=0$

For any two sets A and B, prove that $\text{(A}-\text{B)}\cup(\text{B}\cap\text{A})=\text{A}$

Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas. $\frac{\text{x}^2}{16}-\frac{\text{y}^2}{9}=1$

For the function$\text{f}(\text{x})=\frac{\text{x}^{100}}{100}+\frac{\text{x}^{99}}{99}+...+\frac{\text{x}^2}{2}+\text{x}+1.$
Prove that $\text{f}'(1)=100$$\text{f}'(0)$.

Convert the complex numbers given in Exercises in the polar form:

Find the mean and variance for the following frequency distribution

Class	0-30	30-60	60-90	90-120	120-150	150-180	180-210
Frequencies	2	3	5	10	3	5	2

calculate the mean deviation about median of the following frequency distribution:

$x_i$	5	7	9	11	13	15	17
$f_i$	2	4	6	8	10	12	8

Solve the system of inequality graphically: 3x + 2y $\le$ 150, x + 4y $\le$ 80, x $\le$ 15, y $\ge$ 0, x $\ge$ 0

Find the equation of the circle passing through the points $(4, 1)$ and $(6, 5)$ and whose centre is on the line $4x + y = 16$.