11TH•Mathematics•Complex Numbers•Question 3

(iii)

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

Given $|z|=7$ and $\arg(z)=\dfrac{23\pi}{12}$ .

z=7\left(\cos\frac{23\pi}{12}+i\sin\frac{23\pi}{12}\right)

\frac{23\pi}{12}=2\pi-\frac{\pi}{12}

\cos\frac{23\pi}{12}=\frac{\sqrt{6}+\sqrt{2}}{4},\quad \sin\frac{23\pi}{12}=-\frac{\sqrt{6}-\sqrt{2}}{4}

\boxed{z=\frac{7(\sqrt{6}+\sqrt{2})}{4}-\frac{7(\sqrt{6}-\sqrt{2})}{4}i}

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