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 θ_n + i sin θ_n) = cos (θ₁ + θ₂ + θ₃ + ……. θ_n) + i sin (θ₁ + θ₂ + θ₃ + ……. θ_n)