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Exercise: Polar Representation · Q21

Q.Express z=−2+23 iz=-2+2\sqrt3\,i in polar form and state its modulus and argument.

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For z=−2+23 iz=-2+2\sqrt3\,i: ∣z∣=(−2)2+(23)2=4+12=16=4|z|=\sqrt{(-2)^2+(2\sqrt3)^2}=\sqrt{4+12}=\sqrt{16}=4. Since a=−2<0, b=23>0a=-2<0,\ b=2\sqrt3>0, zz lies in Quadrant II. Reference angle: tan⁡−1(232)=tan⁡−1(3)=π3\tan^{-1}\left(\dfrac{2\sqrt3}{2}\right)=\tan^{-1}(\sqrt3)=\dfrac{\pi}{3}. Quadrant-II argument: θ=π−π3=2π3\theta=\pi-\dfrac{\pi}3=\dfrac{2\pi}{3}. Check: $4\left(\cos\dfrac{2\pi}3+i\sin\dfrac{2\pi}3\right)=4\left(-\dfrac12+i …

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