The complex number $z=\frac{i-1}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$ is equal to:

(A) $\sqrt{2}\left(\cos \frac{5 \pi}{12}+i \sin \frac{5 \pi}{12}\right)$

(B) $\cos \frac{\pi}{12}-\mathrm{i} \sin \frac{\pi}{12}$

(C) $\sqrt{2}\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}\right)$

(D) $\sqrt{2} i\left(\cos \frac{5 \pi}{12}-i \sin \frac{5 \pi}{12}\right)$

JEE Main · 2023 · 31 Jan 2023 · Shift 2 · Complex Numbers

The complex number

z = \frac{i - 1}{c o s \frac{π}{3} + i s i n \frac{π}{3}}

is equal to:

Answer: (A)

Here

i - 1 = 2 (cos \frac{3 π}{4} + i sin \frac{3 π}{4})

, and the denominator

cos \frac{π}{3} + i sin \frac{π}{3}

has modulus

1.

Dividing subtracts the arguments, so

z = 2 (cos (\frac{3 π}{4} - \frac{π}{3}) + i sin (\frac{3 π}{4} - \frac{π}{3})) = 2 (cos \frac{5 π}{12} + i sin \frac{5 π}{12}) .

JEE Main · 2023 · 31 Jan 2023 · Shift 2 · Single correct option