If z is a complex number of unit modulus and argument θ , then arg 1 + z 1 + z - can be equal to given z ≠ - 1

Question

If z is a complex number of unit modulus and argument θ , then arg 1 + z 1 + z - can be equal to given z ≠ - 1, θ, π - θ, - θ, π 2 - θ

MARKS App · Accepted Answer

Let z = cos θ + i sin θ Then, 1 + z 1 + z ¯ = 1 + cos θ + i sin θ 1 + cos θ - i sin θ = 2 cos 2 θ 2 + i · 2 sin θ 2 cos θ 2 2 cos 2 θ 2 - i · 2 sin θ 2 cos θ 2 = cos θ 2 + i sin θ 2 cos θ 2 - i sin θ 2 = cos θ 2 + i sin θ 2 2 = cos θ + i sin θ ∴ a r g 1 + z 1 + z ¯ = a r g z = θ

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