Cpmplex number & Argand diagram?

Question

Given that z = 1 + cos θ + i sin θ.
where -π < θ < π

1. Show that for all values of θ, the point representing z in a Argand diagram is located on a circle. Find the center and the radius of the circle.

2. Prove that the real part of (1/z) is 1/2 for all value of θ.
* I have calculated 1/z and it's correct. 1/z = 1/2 - [i sqrt(3)]/6

Answer 1

1)
let me write t for theta
z = 1+ cos t + i sin t

z = x+iy

gives me

x= 1 + cost ...1
or cos t = x-1 ...2

y = sin t....3

from 2 and 3
(x-1)^2 + y^2 = cos^2 t + sin^2 t = 1

so it is a circle with centre (1,0) radius 1

now 1/z = 1/(1+ cos t) + i sin t)

multiply num and den by conjugate of den

= (1+ cos t) - i sin t)/(1+cos t) + i sin t)(1+ cost - i sin t)

den = (1+cost)^2 + sin^2 t
= (1+ cos^2 t + 2cos t + sin^2 t)
= (1+ 1 + 2 cos t)
= 2(1+ cos t)

real part = (1+cos t)/2(1+cost ) = 1/2

proved

imaginary part = - sin t/(2(1+cos t) )

Answer 2

1) z-1 = cos θ + i sin θ
so magnitude(z-1) = sqrt(cos^2+sin^2) = 1
thus, the diagram is a circle centered at (1,0) and having a
radius of 1.

2) 1/z = 1/(1 + cos θ + i sin θ) =
(1 + cos θ - i sin θ)/[(1+cos)^2+sin^2] =
but
(1+cos)^2+sin^2 = 1+cos^2+2cos + sin^2 = 2+2cos = 2(1+cos)

so 1/z = (1+cos) / [2(1+cos)]- i*sin[2(1+cos)]
so real part of 1/z is (1+cos)/[2(1+cos)]
or it is 1/2.

Answer 3

The real part goes from 0 to 2, the imaginary part goes from -i to i

Thus it's a circle whose center is at (1,0). The radius is 1. Notice that ||z-1|| = 1, the radius.