Mod Amplitude Form (or) Polar Form
Mod Amplitude Form (or) Polar Form: Let z = a+ ib be a complex number such that |z| = r and θ be the amplitude of z. then cosθ = a/ r, sinθ = b/ r.
Now z = a + ib = r cosθ + i r sinθ
= r (cosθ + isinθ)
This know as mod (Modulus) amplitude form or polar form of z
Key point:
⇒ Cosθ + I sinθ is simply denoted by cisθ
⇒ Cosθ + isinθ = eiθ is known as Euler’s formula
⇒ r₁ cisθ₁ = r₂ cisθ₂ ⇔ r₁ = r₂, θ₁ = 2kπ + θ₂, k ϵ Z
⇒ If z₁ = r₁ cisθ₁, z₂ = cisθ₂ then
⇝ z₁z₂ = cis (θ₁ + θ₂)
⇝ z₁/z₂ = r₁/r₂ cis (θ₁ – θ₂)
Example: \({{(\sqrt{3}+i)}^{100}}={{2}^{99}}(a+ib)\), show that a² + b² = 4.
Solution: Given that \({{(\sqrt{3}+i)}^{100}}={{2}^{99}}(a+ib)\),
\(|{{(\sqrt{3}+i)}^{100}}|=|{{2}^{99}}(a+ib)|\),
\({{2}^{100}}={{2}^{99}}\sqrt{{{a}^{2}}+{{b}^{2}}}\),
\(\frac{{{2}^{100}}}{{{2}^{99}}}=\sqrt{{{a}^{2}}+{{b}^{2}}}\),
\(2=\sqrt{{{a}^{2}}+{{b}^{2}}}\),
Squaring on both sides
a² + b² = 4.