# Argand Plane and Polar Representation

The polar form of the complex number z = x + iy is r(cos θ + i sin θ), where r = \sqrt{x^2+y^2} (the modulus of z) and \cos \theta =\frac{x}{r},\sin \theta =\frac{y}{r}. (θ is known as the argument of z. The value of θ, such that −π < θ ≤ π, is called the principal argument of z.

Tips:

• The unique value of ‘θ’ such that –π < θ ≤ π for which x = r.cosθ and y = r.sinθ, is know as the principal value of the argument.
• The general value of the argument is (2nπ + θ) where ‘n’ is an integer and ‘θ’ is the principal value of arg (z).
• While reducing a complex number to polar form, we always take the principle value.
• The complex number z = r(cosθ + i sinθ) can also be written as r.cisθ
• \tt arg \left(\frac{z}{\overline{z}}\right) = 2.arg \ z
• arg(zn) = n.arg(z)
• If \tt arg \left(\frac{z_{2}}{z_{1}}\right) = \theta, then \tt arg \left(\frac{z_{1}}{z_{2}}\right) = 2K\pi - \theta where K∈I
• \tt arg \ \overline{z} = -arg \ z
• arg (positive real number) = 0
• \tt \cos \theta = \frac{e^{i \theta} + e^{-i \theta}}{2} \ and \ \sin \theta = \frac{e^{i \theta} - e^{-i \theta}}{2i}

Tricks:

• If x > 0, y > 0 then \tt arg \ z = \theta = Tan^{-1}\left(\frac{y}{x}\right)
• If x < 0, y > 0 then \tt arg \ z = \theta = \pi - Tan^{-1} \frac{y}{|x|}
• If x < 0, y < 0 then \tt arg \ z = \theta = -\pi + Tan^{-1} \frac{y}{x}
• If x > 0, y < 0 then \tt arg \ z = \theta = -Tan^{-1} \frac{|y|}{x}
• Argument of the complex number ‘o’ is not defined
• Arg (x + i0) = 0 if x > 0
= π if x < 0
• Arg (0 + iy) = π/2 if y > 0
= 3π/2 if y < 0
• If ‘n’ is any rational number, then cos nθ + I sin nθ is one of the value of (cos θ + I sin θ)n
• (cos θ + i sin θ)−n = cos(−n) θ + i sin(−n)θ = cos n θ – i sin nθ
• (cos θ – i sin θ)n = cosnθ − i sin nθ
• \tt \frac{1}{\cos \theta + i \sin \theta} = \cos \theta - i \sin \theta
• The theorem cannot be applied for (cos θ + I sin φ)n i.e ‘θ’ and φ are different
• (sin θ + i cos θ)n = \tt \cos n \left(\frac{\pi}{2} - \theta\right) + i \sin n \left(\frac{\pi}{2} - \theta\right)
• (cos θ1 + i sin θ1) (cos θ2 + i sin θ2)……. (cos θn + i sin θn) = cos (θ1 + θ2 + θ3 + ……. θn) + i sin (θ1 + θ2 + θ3 + ……. θn)

### part2: View the Topic in this video From 00:40 To 20:46

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Properties of Argument
a) arg \tt \left(\overline{z}\right)=-arg\left(z\right)
b) arg (z1 z2) = arg (z1) + arg (z2) + 2 kπ (k = 0, 1 or −1)
In general,
arg(z1z2z3 ...... zn) = arg (z1) + arg (z2) + arg (z3) + ....+ arg (zn) + 2kπ (k = 0, 1 or −1)
c) \tt arg\left(\frac{z_{1}}{z_{2}}   \right) = arg (z1) − arg (z2) + 2kπ (k = 0, 1 or = −1)
d) \tt arg\left(z_{1}\overline{z_{2}}\right)= arg (z1) - arg (z2)
e) \tt arg\left(\frac{z}{\overline{z}}\right) = 2 arg (z) + 2kπ (k = 0, 1 or −1)
f) arg (zn) = n arg (z) + 2kπ (k = 0, 1 or −1)
g) If \tt arg\left(\frac{z_{2}}{z_{1}}\right)=\theta, then\ arg\left(\frac{z_{1}}{z_{2}}\right)=2k\pi-\theta, k\epsilon I
h) If arg (z) = 0 ⇒ z is real
i) arg (z) - arg (-z) = \tt \begin{cases}\pi, if \ arg \ \left(z\right)> 0 \\ \tt -\pi, if \ arg \ \left(z\right) <0 & \end{cases}
j) If |z1 + z2| = |z1 − z2| , then arg \tt \left(\frac{z_{1}}{z_{2}}\right)=arg \left(z_{1}\right)-arg\left(z_{2}\right)=\frac{\pi}{2}
k) If |z1 + z2|= |z1| + |z2|, then arg (z1) = arg (z2)
l) If |z − 1| = |z + 1|, then arg (z) = ± \tt \frac{\pi}{2}
m) If arg \tt \left(\frac{z-1}{z+2}\right)=\frac{\pi}{2}, then\left(z\right)=1
n) If arg \tt \left(\frac{z+1}{z-2}\right)=\frac{\pi}{2}, then z lies on circle of radius unity and centre at origin.