Advertisements
Advertisements
Question
Find principal argument of `(1 + i sqrt(3))^2`.
Advertisements
Solution
Given that: `(1 + i sqrt(3))^2 = 1 + i^2 . 3 + 2sqrt(3) i`
= `1 - 3 + 2sqrt(3)i`
= `-2 + 2sqrt(3)i`
`tan alpha = |(2sqrt(3))/2|` ......`[because tan alpha = |("Img"(z))/("Re"(z))|]`
⇒ `tan alpha = |- sqrt(3)| = sqrt(3)`
⇒ `tan alpha = tan pi/3`
∴ `alpha = pi/3`
Now Re(z) < 0 and image(z) > 0.
∴ arg(z) = `pi - alpha`
= `pi - pi/3`
= `(2pi)/3`
Hence, the principal arg = `(2pi)/3`.
APPEARS IN
RELATED QUESTIONS
Find the modulus and the argument of the complex number `z =- sqrt3 + i`
Convert the given complex number in polar form: – 1 + i
Convert the given complex number in polar form: – 1 – i
Convert the given complex number in polar form: –3
Convert the given complex number in polar form: i
Convert the following in the polar form:
`(1+7i)/(2-i)^2`
Convert the following in the polar form:
`(1+3i)/(1-2i)`
If the imaginary part of `(2z + 1)/(iz + 1)` is –2, then show that the locus of the point representing z in the argand plane is a straight line.
Let z1 and z2 be two complex numbers such that `barz_1 + ibarz_2` = 0 and arg(z1 z2) = π. Then find arg (z1).
If |z| = 2 and arg(z) = `pi/4`, then z = ______.
The locus of z satisfying arg(z) = `pi/3` is ______.
What is the polar form of the complex number (i25)3?
The amplitude of `sin pi/5 + i(1 - cos pi/5)` is ______.
Show that the complex number z, satisfying the condition arg`((z - 1)/(z + 1)) = pi/4` lies on a circle.
If arg(z – 1) = arg(z + 3i), then find x – 1 : y. where z = x + iy.
z1 and z2 are two complex numbers such that |z1| = |z2| and arg(z1) + arg(z2) = π, then show that z1 = `-barz_2`.
If for complex numbers z1 and z2, arg (z1) – arg (z2) = 0, then show that `|z_1 - z_2| = |z_1| - |z_2|`.
If z and w are two complex numbers such that |zw| = 1 and arg(z) – arg(w) = `pi/2`, then show that `barz`w = –i.
arg(z) + arg`barz (barz ≠ 0)` is ______.
If |z| = 4 and arg(z) = `(5pi)/6`, then z = ______.
State True or False for the following:
Let z1 and z2 be two complex numbers such that |z1 + z2| = |z1| + |z2|, then arg(z1 – z2) = 0.
Find z if |z| = 4 and arg(z) = `(5pi)/6`.
The value of arg (x) when x < 0 is ______.
If arg(z) < 0, then arg(–z) – arg(z) = ______.
