Write the polar form of z=1+cosa+i*sina.

The complex number z can be represented by the plane polar
coordinates (r, a).

z = r(cos theta+ i*sin
theta)

We know, from enunciation, the original form of the
given complex 
number:

z=1+cosa+i*sina

Re(z)
= 1 + cos a

Im (z) = sin
a

We'll write 1 + cos a = 2 [cos
(a/2)]^2

sin a = 2 sin(a/2)*cos
(a/2)

We'll re-write z:

z = 2
[cos (a/2)]^2 + i*2 sin(a/2)*cos (a/2)

We'll factorize by
2cos (a/2):

z = 2cos (a/2)[cos (a/2)
+ sin(a/2)*i]

Now, we'll identify the polar coordinates r
and a:

r = 2cos (a/2)

theta =
arg (z) = (a/2)

The polar form of z
is:

z = 2cos (a/2)[cos (a/2)
+ sin(a/2)*i]