4.4 Argand Plane and Polar Representation

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  1. Modulus: r=∣z∣=(−1)2+(−3)2=1+3=2r = |z| = \sqrt{(-1)^2 + (-\sqrt{3})^2} = \sqrt{1+3} = 2.
  2. Alpha: α=tan⁡−1∣−3−1∣=tan⁡−13=π3\alpha = \tan^{-1}|\frac{-\sqrt{3}}{-1}| = \tan^{-1}\sqrt{3} = \frac{\pi}{3}.
  3. Quadrant: Since x<0,y<0x<0, y<0, it is III Quadrant.
  4. Argument: θ=−(π−π/3)=−2π3\theta = -(\pi - \pi/3) = -\frac{2\pi}{3}.
  5. Polar Form: z=2(cos⁡(−2π3)+isin⁡(−2π3))=2(cos⁡2π3−isin⁡2π3)z = 2(\cos(-\frac{2\pi}{3}) + i\sin(-\frac{2\pi}{3})) = 2(\cos \frac{2\pi}{3} - i \sin \frac{2\pi}{3}).

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