If `z=r e^(itheta)` , then prove that `|e^(i z)|=e^(-r sintheta)`

If z= 5e^(itheta), then |e^(iz)| is equal to

If z=re^(i)theta then |e^(iz)| is equal to:

In argand plane |z| represent the distance of a point z from the origin. In general |z_(1)-z_(2)| represent the distance between two points z_(1) and z_(2) . Also for a general moving point z in argand plane, if arg(z)=theta , then z=|z|e^(i theta) , where e^(i theta)=cos theta+i sin theta . If z_(1)=4e^(i pi//3) and z_(2)=2e^(i5pi//6) , then |z_(1)-z_(2)| equals

The value of |e^(z)|=

In argand plane |z| represent the distance of a point z from the origin. In general |z_(1)-z_(2)| represent the distance between two points z_(1) and z_(2) . Also for a general moving point z in argand plane, if arg(z)=theta , then z=|z|e^(i theta) , where e^(i theta)=cos theta+i sin theta . If |z-(3+2i)|=|z cos((pi)/(4)-"arg z")| , then locus of z is

Read the following writeup carefully: In argand plane |z| represent the distance of a point z from the origin. In general |z_1-z_2| represent the distance between two points z_1 and z_2 . Also for a general moving point z in argand plane, if arg(z) =theta , then z=|z|e^(itheta) , where e^(itheta) = cos theta + i sintheta . Now answer the following question If |z-(3+2i)|=|z cos ((pi)/(4) - "arg" z)|, then locus of z is