If `f(theta)= |[1,1,-1] , [1,e^(itheta),1] , [1,-1,-e^(-itheta)] |`

(1 + e^(-i theta))/(1 + e^(i theta)) =

The value of e^(I theta) - e^(-itheta) 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

let z1, z2,z3 be vertices of triangle ABC in an anticlockwise order and angle ACB = theta then z_2-z_3 = (CB)/(CA)(z_1-z3) e^itheta . let p point on a circle with op diameter 2 points Q & R taken on a circle such that angle POQ & QOR= theta if O be origin and PQR are complex no. z1, z2, z3 respectively then z_2/z_1 = (A) e^(itheta) cos theta (B) e^(itheta) cos 2theta (C) e^(-itheta) cos theta (D) e^(2itheta) cos 2theta

z_2/z_1 = (A) e^(itheta) cos theta (B) e^(itheta) cos 2theta (C) e^(-itheta) cos theta (D) e^(2itheta) cos 2theta

If z_1 and z_2 are two complex numbers for which |(z_1-z_2)(1-z_1z_2)|=1 and |z_2|!=1 then (A) |z_2|=2 (B) |z_1|=1 (C) z_1=e^(itheta) (D) z_2=e^(itheta)

If z_1 and z_2 are two complex numbers for which |(z_1-z_2)(1-z_1z_2)|=1 and |z_2|!=1 then (A) |z_2|=2 (B) |z_1|=1 (C) z_1=e^(itheta) (D) z_2=e^(itheta)

Let P(e^(itheta_1)), Q(e^(itheta_2)) and R(e^(itheta_3)) be the vertices of a triangle PQR in the Argand Plane. Theorthocenter of the triangle PQR is

Real part of e^e^(itheta) is