The principal value of the arg (z) and | z | of the complex number z=1+ cos((11π/9))+ i sin(11π/9) are respectively

Question

The principal value of the arg (z) and | z | of the complex number z=1+ cos((11π/9))+ i sin(11π/9) are respectively (A) (11π/8),2 cos((π/18))  (B) -(7π/18),-2 cos((11π/18))  (C) (2π/9),2 cos((7π/18))  (D) -(π/9),-2 cos((π/18))

Tardigrade Admin · Accepted Answer

z=2 cos2 (11π/18)+2i sin (11π/18) cos (11π/18) =2 cos (11π/18) cis ((11π/18)) But (11π/18) is in the Ilnd quadrant ∴ cos (11π/18) < 0 ∴ z=-2 cos((11π/18))cis((11π/18)-π) = -2 cos((11π/18))cis(-(7π/18)) ∴ Arg z=-(7π/18) i.e., |z|=-2 cos((π/18))

Q. The principal value of the $a r g (z)$ and $∣ z ∣$ of the complex number $z = 1 + cos (\frac{11 π}{9}) + i sin \frac{11 π}{9}$ are respectively

$\frac{11 π}{8}, 2 cos (\frac{π}{18})$

$- \frac{7 π}{18}, - 2 cos (\frac{11 π}{18})$

$\frac{2 π}{9}, 2 cos (\frac{7 π}{18})$

$- \frac{π}{9}, - 2 cos (\frac{π}{18})$

Q. The principal value of the arg(z) and ∣z∣ of the complex number z=1+cos(911π​)+isin911π​ are respectively

811π​,2cos(18π​)

−187π​,−2cos(1811π​)

92π​,2cos(187π​)

−9π​,−2cos(18π​)

z=2cos21811π​+2isin1811π​cos1811π​ =2cos1811π​cis(1811π​) But 1811π​ is in the Ilnd quadrant ∴cos1811π​<0 ∴z=−2cos(1811π​)cis(1811π​−π) =−2cos(1811π​)cis(−187π​) ∴Argz=−187π​ i.e., ∣z∣=−2cos(18π​)

Q. The principal value of the $a r g (z)$ and $∣ z ∣$ of the complex number $z = 1 + cos (\frac{11 π}{9}) + i sin \frac{11 π}{9}$ are respectively

$\frac{11 π}{8}, 2 cos (\frac{π}{18})$

$- \frac{7 π}{18}, - 2 cos (\frac{11 π}{18})$

$\frac{2 π}{9}, 2 cos (\frac{7 π}{18})$

$- \frac{π}{9}, - 2 cos (\frac{π}{18})$