The complex number z=i-1 ( )/(3)+i ( )/(3) is equal to :

Question

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

Accepted Answer

$$z=\frac{i-1}{\frac12+\frac{\sqrt3}2i}=\frac{(i-1)\left(\frac12-\frac{\sqrt3}2i\right)}{\left(\frac12\right)^2+\left(\frac{\sqrt3}2\right)^2}=(i-1)\left(\frac12-\frac{\sqrt3}2i\right)=\frac{\sqrt3-1}2+\frac{\sqrt3+1}2i.$$

Writing $z=r(\cos\theta+i\sin\theta)$: $r\cos\theta=\dfrac{\sqrt3-1}2,\ r\sin\theta=\dfrac{\sqrt3+1}2$.

$$r^2=\left(\frac{\sqrt3-1}2\right)^2+\left(\frac{\sqrt3+1}2\right)^2=\frac{(4-2\sqrt3)+(4+2\sqrt3)}4=2 \implies r=\sqrt2.$$

$$\tan\theta=\frac{\sqrt3+1}{\sqrt3-1} \implies \theta=\frac{5\pi}{12}.$$

$$z=\sqrt2\left(\cos\frac{5\pi}{12}+i\sin\frac{5\pi}{12}\right).$$

Solutions · 1