Find the modulus and argument of the complex number 1+i 1-i - Vidyadip

Question

Find the modulus and argument of the complex number $\frac{1+i}{1-i}$

✓

Answer

We have $\frac{1+i}{1-i}=\frac{1+i}{1-i} \times \frac{1+i}{1+i}=\frac{1-1+2 i}{1+1} = i = 0 + i$
Now, let us put $0 = r \cos \theta , 1 = r \sin \theta$
Squaring and adding, $r^2 = 1$ i.e., $r = 1$ so that
$\cos \theta = 0, \sin \theta = 1$
Therefore, $\theta = \frac {\pi}{2}$
Hence, the modulus of $\frac {1+i}{1-i}$ is $1$ and the argument is $\frac {\pi}2.$

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 "3 Marks Question" from Complex Numbers and Quadratic Equations→Complex Numbers and Quadratic Equations — all question types→Maths STD 11 Science question papers→CBSE Board STD 11 Science question papers→CBSE Board question paper generator→

Similar questions

Expand the given expression $(1 - 2x)^5$

If the $4^{th}, 10^{th} $ and $16^{th} $ terms of a G.P. are $x, y$ and $z$ respectively. Prove that $x, y , z$ are in G.P.

Find the equation of a parabola with vertex at the origin and the directrix, $y = 2.$

Find the real numbers x and y if (x - iy) (3+ 5i) is the conjugate of -6 - 24i.

Evaluate the following limits:
$\lim\limits_{\text{x}\rightarrow\infty}\frac{3\text{x}^{-1}+4\text{x}^{-2}}{5\text{x}^{-1}+6\text{x}^{-2}}$

Find the sum of the following series to n terms:
$2^2 + 4^2 + 6^2 + 8^2 + ....$

Evaluate:
$\lim\limits_{\text{x} \rightarrow{\text{a}}}\frac{(2+\text{x})^\frac{5}{2}-(\text{a}+2)^\frac{5}{2}}{\text{x}-\text{a}}$

Prove that:
$\big(\cos\alpha+\cos\beta^2\big)+\big(\sin\alpha+\sin\beta\big)^2=2\cos^2\Big(\frac{\alpha-\beta}{2}\Big)$

Find the real values of x and y, if
$(3\text{x}-2\text{iy})(2+\text{i})^2=10(1+\text{i})$

Find the coefficient of $x^{15}$ in the expansion of $(x - x^2)^{10}.$