Answer to Question #262542 in Complex Analysis for luka

Question #262542

Determine the modulus and argument of z1=1+cos θ+isin θ/1-cos θ+isin θ


1
Expert's answer
2021-11-09T13:24:00-0500
1+cosθ+isinθ1cosθ+isinθ\dfrac{1+\cos θ+i\sin θ}{1-\cos θ+i\sin θ}

=2cos2(θ2)+2sin(θ2)cos(θ2)i2sin2(θ2)+2sin(θ2)cos(θ2)i=\dfrac{2\cos^2(\dfrac{\theta}{2})+2\sin(\dfrac{\theta}{2})\cos(\dfrac{\theta}{2})i}{2\sin^2(\dfrac{\theta}{2})+2\sin(\dfrac{\theta}{2})\cos(\dfrac{\theta}{2})i}

=2cos(θ2)(cos(θ2)+sin(θ2)i)2sin(θ2)(cos(θ2)+sin(θ2)i)=\dfrac{2\cos(\dfrac{\theta}{2})\big(\cos(\dfrac{\theta}{2})+\sin(\dfrac{\theta}{2})i\big)}{2\sin(\dfrac{\theta}{2})\big(\cos(\dfrac{\theta}{2})+\sin(\dfrac{\theta}{2})i\big)}

=cot(θ2),sin(θ2)0=\cot(\dfrac{\theta}{2}), \sin(\dfrac{\theta}{2})\not=0

z=cot(θ2),Arg(z)=0,θπn,nZ|z|=\cot(\dfrac{\theta}{2}), Arg(z)=0, \theta\not=\pi n, n\in\Z


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