The value of z₁ - z₂ is C- cos(pi)+isin(pi)
Since z₁ = √2[cos(3π/4) + isin(3π/4)] and z₂ = [cos(π/2) + isin(π/2)]
z₁ - z₂ = √2[cos(3π/4) + isin(3π/4)] - ([cos(π/2) + isin(π/2)])
= √2[cos(3π/4)] + i√2[sin(3π/4)] - cos(π/2) - isin(π/2)
= √2[cos(3π/4)] - cos(π/2) + i√2[sin(3π/4)] - isin(π/2)
Since 3π/4 = 3π/4 × 180°/π = 135° and π/2 = π/2 × 180°/π = 90°,
So,
z₁ - z₂ = √2[cos(3π/4)] - cos(π/2) + i√2[sin(3π/4)] - isin(π/2)
z₁ - z₂ = √2[cos135°] - cos90° + i√2[sin135°] - isin90°
z₁ - z₂ = √2[cos(180° - 45°) - cos90° + i(√2[sin(180° - 45°)] - sin90°)
z₁ - z₂ = √2[-cos45°] - cos90° + i√2[sin45°] - isin90°
cos45° = sin45° = 1/√2 and sin90° = 1 and cos90° = 0.
So,
z₁ - z₂ = √2[-1/√2 ] - 0 + i(√2[1/√2 ] - 1)
z₁ - z₂ = - 1 - 0 + i(1 - 1)
z₁ - z₂ = - 1 + i0
z₁ - z₂ = - 1
So, the magnitude of |z₁ - z₂| = |- 1| = 1 = r
It's argument tanθ = sinθ/cosθ, since sinθ = 0 and cosθ = -1
tanθ = 0/-1 = -0
tanθ = -0 ⇒ tan(π - θ) = 0
Taking inverse tan of both sides, we have
π - θ = tan⁻¹(0)
π - θ = 0
π = θ
Since z₁ - z₂ = r[cosθ + isinθ] and r = 1 and θ = π, then
z₁ - z₂ = r[cosθ + isinθ]
z₁ - z₂ = 1[cosπ + isinπ]
z₁ - z₂ = cosπ + isinπ
The value of z₁ - z₂ is C- cos(pi)+isin(pi)
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