Advertisements
Advertisements
Question
Find the modulus and argument of the following complex number and hence express in the polar form:
\[\sqrt{3} + i\]
Solution
\[ z = \sqrt{3} + i\]
\[r = \left| z \right|\]
\[ = \sqrt{3 + 1}\]
\[ = \sqrt{4}\]
\[ = 2\]
\[\text { Let } \tan \alpha = \left| \frac{Im\left( z \right)}{Re\left( z \right)} \right|\]
\[ \Rightarrow \tan \alpha = \left( \frac{1}{\sqrt{3}} \right)\]
\[ \Rightarrow \alpha = \frac{\pi}{6}\]
\[\text { Since point } (\sqrt{3}, 1) \text { lies in the first quadrant, the argument of z is given by } \]
\[\theta = \alpha = \frac{\pi}{6}\]
\[\text { Polar form } = r \left( \cos\theta + i\sin\theta \right) \]
\[ = 2 \left( \cos \frac{\pi}{6} + i\sin\frac{\pi}{6} \right)\]
APPEARS IN
RELATED QUESTIONS
Find the modulus and argument of the complex number `(1 + 2i)/(1-3i)`
Find the real numbers x and y if (x – iy) (3 + 5i) is the conjugate of –6 – 24i.
Find the conjugate of the following complex number:
4 − 5 i
Find the conjugate of the following complex number:
\[\frac{1}{3 + 5i}\]
Find the conjugate of the following complex number:
\[\frac{(3 - i )^2}{2 + i}\]
Find the conjugate of the following complex number:
\[\frac{(1 + i)(2 + i)}{3 + i}\]
Find the modulus of \[\frac{1 + i}{1 - i} - \frac{1 - i}{1 + i}\].
Find the modulus and argument of the following complex number and hence express in the polar form:
1 + i
Find the modulus and argument of the following complex number and hence express in the polar form:
1 − i
Find the modulus and argument of the following complex number and hence express in the polar form:
\[\frac{1 - i}{1 + i}\]
Find the modulus and argument of the following complex number and hence express in the polar form:
\[\frac{1}{1 + i}\]
Find the modulus and argument of the following complex number and hence express in the polar form:
\[\frac{1 + 2i}{1 - 3i}\]
Find the modulus and argument of the following complex number and hence express in the polar form:
\[\frac{- 16}{1 + i\sqrt{3}}\]
If z1, z2 and z3, z4 are two pairs of conjugate complex numbers, prove that \[\arg\left( \frac{z_1}{z_4} \right) + \arg\left( \frac{z_2}{z_3} \right) = 0\].
Write the conjugate of \[\frac{2 - i}{\left( 1 - 2i \right)^2}\] .
If (1+i)(1 + 2i)(1+3i)..... (1+ ni) = a+ib,then 2 ×5 ×10 ×...... ×(1+n2) is equal to.
If (1 + i) (1 + 2i) (1 + 3i) .... (1 + ni) = a + ib, then 2.5.10.17.......(1+n2)=
If \[x + iy = (1 + i)(1 + 2i)(1 + 3i)\],then x2 + y2 =
If \[\frac{1 - ix}{1 + ix} = a + ib\] then \[a^2 + b^2\]
If |z2 – 1| = |z|2 + 1, then show that z lies on imaginary axis.
The conjugate of the complex number `(1 - i)/(1 + i)` is ______.
If z1 = `sqrt(3) + i sqrt(3)` and z2 = `sqrt(3) + i`, then find the quadrant in which `(z_1/z_2)` lies.
What is the conjugate of `(sqrt(5 + 12i) + sqrt(5 - 12i))/(sqrt(5 + 12i) - sqrt(5 - 12i))`?
If z1, z2 and z3, z4 are two pairs of conjugate complex numbers, then find arg`(z_1/z_4)` + arg`(z_2/z_3)`.
State True or False for the following:
If z is a complex number such that z ≠ 0 and Re(z) = 0, then Im(z2) = 0.
What is the conjugate of `(2 - i)/(1 - 2i)^2`?
sinx + icos2x and cosx – isin2x are conjugate to each other for ______.
If z = x + iy lies in the third quadrant, then `barz/z` also lies in the third quadrant if ______.
