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

(iv) $-\dfrac{5}{2}-\dfrac{5\sqrt{3}}{2}i$

Express the following complex numbers in polar form:

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(iv) −52−532i-\dfrac{5}{2}-\dfrac{5\sqrt{3}}{2}i

r=(−52)2+(−532)2=5 r=\sqrt{\left(-\frac{5}{2}\right)^2+\left(-\frac{5\sqrt{3}}{2}\right)^2}=5

Both x<0x<0 and y<0y<0, so the point lies in Quadrant III.

tan⁡−1(yx)=tan⁡−1(3)=π3⇒θ=π3+π=4π3\tan^{-1}\left(\frac{y}{x}\right)=\tan^{-1}(\sqrt{3})=\frac{\pi}{3} \Rightarrow \theta=\frac{\pi}{3}+\pi=\frac{4\pi}{3} −52−532i=5(cos⁡4π3+isin⁡4π3)\boxed{-\frac{5}{2}-\frac{5\sqrt{3}}{2}i=5\left(\cos\frac{4\pi}{3}+i\sin\frac{4\pi}{3}\right)}

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