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Mathematics Question on complex numbers

The complex number z=i−1cos⁡π3+isin⁡π3z=\frac{i-1}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}} is equal to :

A

2i(cos⁡5π12−isin⁡5π12)\sqrt{2} i\left(\cos \frac{5 \pi}{12}-i \sin \frac{5 \pi}{12}\right)

B

cos⁡π12−isin⁡π12\cos \frac{\pi}{12}-i \sin \frac{\pi}{12}

C

2(cos⁡π12+isin⁡π12)\sqrt{2}\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}\right)

D

2(cos⁡5π12+isin⁡5π12)\sqrt{2}\left(\cos \frac{5 \pi}{12}+i \sin \frac{5 \pi}{12}\right)

Answer

2(cos⁡5π12+isin⁡5π12)\sqrt{2}\left(\cos \frac{5 \pi}{12}+i \sin \frac{5 \pi}{12}\right)

Explanation

Solution

Z=cos3π​+isin3π​i−1​=21​+23​​ii−1​
=21​+23​​ii−1​×21​−3/2​i21​−23​i​​=23​−1​+23​+1​i
Apply polar form,
rcosθ=23​−1​
rsinθ=23​+1​
Now, tanθ=3​−13​+1​
So, θ=125π​
So, the correct option is (D) : 2(cos⁡5π12+isin⁡5π12)\sqrt{2}\left(\cos \frac{5 \pi}{12}+i \sin \frac{5 \pi}{12}\right)