Question

Find the conjugate of the given complex number

Answer 1

Answer:

Step-by-step explanation:

We have,

$z=\dfrac{i-1}{\cos\left(\dfrac{\pi}{3}\right)+i\,\sin\left(\dfrac{\pi}{3}\right)}$

$\implies\,z=\dfrac{i-1}{\dfrac{1}{2}+i\,\dfrac{\sqrt{3}}{2}}$

$\implies\,z=\dfrac{2(i-1)}{1+i\sqrt{3}}$

$\implies\,z=\dfrac{2(i-1)\left(1-i\sqrt{3}\right)}{\left(1+i\sqrt{3}\right)\left(1-i\sqrt{3}\right)}$

$\implies\,z=\dfrac{2(i-1)\left(1-i\sqrt{3}\right)}{1+(\sqrt{3})^2}$

$\implies\,z=\dfrac{2\left(i-1-i^2\sqrt{3}+i\sqrt{3}\right)}{1+3}$

$\implies\,z=\dfrac{2\left(i-1+\sqrt{3}+i\sqrt{3}\right)}{4}$

$\implies\,z=\dfrac{(-1+\sqrt{3})+(1+\sqrt{3})i}{2}$

$\implies\,z=\left(\dfrac{\sqrt{3}-1}{2}\right)+\left(\dfrac{\sqrt{3}+1}{2}\right)i$

Now,

$\implies\,\bar{z}=\left(\dfrac{\sqrt{3}-1}{2}\right)-\left(\dfrac{\sqrt{3}+1}{2}\right)i$