Theory of Intelligence
11TH•Mathematics•Complex Numbers•Question 4

(iii) $z_1\cdot z_2$

If $z_1=9\left(\cos\dfrac{5\pi}{4}+i\sin\dfrac{5\pi}{4}\right)$ and $z_2=5\left(\cos\dfrac{\pi}{3}+i\sin\dfrac{\pi}{3}\right)$ then find:

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(iii) z1⋅z2z_1\cdot z_2

z1z2=45(cos⁡19π12+isin⁡19π12)z_1z_2=45\left(\cos\frac{19\pi}{12}+i\sin\frac{19\pi}{12}\right) cos⁡19π12=6−24,sin⁡19π12=−6+24\cos\frac{19\pi}{12}=\frac{\sqrt{6}-\sqrt{2}}{4},\quad \sin\frac{19\pi}{12}=-\frac{\sqrt{6}+\sqrt{2}}{4} z1z2=45(6−2)4−45(6+2)4i\boxed{z_1z_2=\frac{45(\sqrt{6}-\sqrt{2})}{4}-\frac{45(\sqrt{6}+\sqrt{2})}{4}i}

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