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

(iii) $\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i$

Express the following complex numbers in polar form:

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(iii) 12+32i\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i

r=(12)2+(32)2=1 r=\sqrt{\left(\frac{1}{2}\right)^2+\left(\frac{\sqrt{3}}{2}\right)^2}=1

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

θ=tan⁡−1(3)=π3\theta=\tan^{-1}(\sqrt{3})=\frac{\pi}{3} 12+32i=cos⁡π3+isin⁡π3\boxed{\frac{1}{2}+\frac{\sqrt{3}}{2}i=\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}}

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