If z=r( cos θ +i sin θ ), then the value of (z/z)=( overlinez/z)

Question

If z=r( cos θ +i sin θ ), then the value of (z/z)=( overlinez/z)  (A)  cos 2θ  (B)  2 cos 2θ  (C)  2 cos θ  (D)  2 sin θ

Tardigrade Admin · Accepted Answer

We have z=r( cos θ +i sin θ ) ∴ overlinez=r( cos θ -i sin θ ) ∴ (z/z)+( overlinez/z) =(r( cos θ +i sin θ )/r( cos θ -i sin θ ))+(r( cos θ -i sin θ )/r( cos θ +i sin θ )) =(( cos θ +i sin θ )2+( cos θ -i sin θ 2)/ cos 2θ + sin 2θ ) = cos 2θ - sin 2θ +2i cos θ sin θ + cos 2θ - sin 2θ +2i cos θ sin θ =2( cos 2θ - sin 2θ ) =2 cos 2θ

Q. If $z = r (cos θ + i sin θ),$ then the value of $\frac{z}{z} = \frac{z}{z}$

$cos 2 θ$

$2 cos 2 θ$

$2 cos θ$

$2 sin θ$

$2 sin 2 θ$

Q. If z=r(cosθ+isinθ), then the value of zz​=zz​

cos2θ

2cos2θ

2cosθ

2sinθ

2sin2θ