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

(iv)

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

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(iv)

Given ∣z∣=11|z|=11 and arg⁡(z)=−11π12\arg(z)=-\dfrac{11\pi}{12}.

z=11(cos⁡(−11π12)+isin⁡(−11π12))z=11\left(\cos\left(-\frac{11\pi}{12}\right)+i\sin\left(-\frac{11\pi}{12}\right)\right) cos⁡(−11π12)=−6+24,sin⁡(−11π12)=−6−24\cos\left(-\frac{11\pi}{12}\right)=-\frac{\sqrt{6}+\sqrt{2}}{4},\quad \sin\left(-\frac{11\pi}{12}\right)=-\frac{\sqrt{6}-\sqrt{2}}{4} z=−11(6+2)4−11(6−2)4i\boxed{z=-\frac{11(\sqrt{6}+\sqrt{2})}{4}-\frac{11(\sqrt{6}-\sqrt{2})}{4}i}

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