Theory of Intelligence
11TH•Mathematics•Complex Numbers•Question 2

(viii) $\dfrac{3}{2}+\dfrac{\sqrt{3}}{2}i$

Express the following complex numbers in polar form:

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(viii) 32+32i\dfrac{3}{2}+\dfrac{\sqrt{3}}{2}i

r=(32)2+(32)2=3 r=\sqrt{\left(\frac{3}{2}\right)^2+\left(\frac{\sqrt{3}}{2}\right)^2}=\sqrt{3}

Since x>0x>0 and y>0y>0:

θ=tan⁡−1(3232)=tan⁡−1(13)=π6\theta=\tan^{-1}\left(\frac{\frac{\sqrt{3}}{2}}{\frac{3}{2}}\right)=\tan^{-1}\left(\frac{1}{\sqrt{3}}\right)=\frac{\pi}{6} 32+32i=3(cos⁡π6+isin⁡π6)\boxed{\frac{3}{2}+\frac{\sqrt{3}}{2}i=\sqrt{3}\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right)}

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