Theory of Intelligence
11TH•Mathematics•Complex Numbers•Question 3

(v)

Convert each of the complex numbers $z$ in the rectangular form $x+iy$:

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(v)

Given ∣z∣=103|z|=\dfrac{10}{3} and arg⁡(z)=−17π12\arg(z)=-\dfrac{17\pi}{12}.

z=103(cos⁡(−17π12)+isin⁡(−17π12))z=\frac{10}{3}\left(\cos\left(-\frac{17\pi}{12}\right)+i\sin\left(-\frac{17\pi}{12}\right)\right) −17π12=−π−5π12-\frac{17\pi}{12}=-\pi-\frac{5\pi}{12} cos⁡(−17π12)=cos⁡(π+5π12)=−cos⁡5π12=2−64\cos\left(-\frac{17\pi}{12}\right)=\cos\left(\pi+\frac{5\pi}{12}\right)=-\cos\frac{5\pi}{12}=\frac{\sqrt{2}-\sqrt{6}}{4} sin⁡(−17π12)=−sin⁡(π+5π12)=sin⁡5π12=6+24\sin\left(-\frac{17\pi}{12}\right)=-\sin\left(\pi+\frac{5\pi}{12}\right)=\sin\frac{5\pi}{12}=\frac{\sqrt{6}+\sqrt{2}}{4} z=5(2−6)6+5(6+2)6i\boxed{z=\frac{5(\sqrt{2}-\sqrt{6})}{6}+\frac{5(\sqrt{6}+\sqrt{2})}{6}i}

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