Help solving complex number Math problem

[denian]

hope to get the idea on how to solve this question.

the complex number z is given by

z = 1 + cos (theta) + i sin (theta)

where -pi < theta < or = +pi


show that for all values of theta, the point representing z in a Argand diagram is located on a circle. find the centre and radius of the circle.

 

[HallsofIvy]

If z = 1 + cos (&theta;) + i sin (&theta)

Then z-1= cos(&theta)+ i sin(&theta;).

If you represent z as x+ iy then
(x-1)+ iy= cos(&theta;)+ i sin(&theta;)

or x- 1= cos(&theta;), y= sin(&theta;)

Those are parametric equations of a circle with what center and radius?

 

[denian]

ok. i compare those with the
y=r sin (&theta)
and
x=r cos (&theta)


so, i know the radius = 1 unit
but may i know how to find the centre of the circle?

tq.

 

[HallsofIvy]

It's exactly where the center of the circle given by

x= r cos &theta;
y= r sin &theta; is!

Hint: x²= r²cos²&theta;
y²= r²sin²&theta;

What is x²+ y²?

If that's too complicated, what is (x,y) when &theta;= 0?
What is (x,y) when &theta;= &pi;?

 

[denian]

tq. u helped me solved the problem.

but there is another part of the question which i ahain need some idea.

--> prove that the real part of (1/z) is (1/2) for all values of [the]