Theory of Intelligence
11TH•Mathematics•Complex Numbers•Question 5

(iii) $z_1\cdot z_2$

If $z_1=7\left(\cos\dfrac{23\pi}{12}+i\sin\dfrac{23\pi}{12}\right)$ and $z_2=11\left(\cos\dfrac{11\pi}{12}+i\sin\dfrac{11\pi}{12}\right)$ then find the following and express the result into $x+iy$ form:

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(iii) z1⋅z2z_1\cdot z_2

z1z2=77(cos⁡5π6+isin⁡5π6)z_1z_2=77\left(\cos\frac{5\pi}{6}+i\sin\frac{5\pi}{6}\right) cos⁡5π6=−32,sin⁡5π6=12\cos\frac{5\pi}{6}=-\frac{\sqrt{3}}{2},\quad \sin\frac{5\pi}{6}=\frac{1}{2} z1z2=−7732+772i\boxed{z_1z_2=-\frac{77\sqrt{3}}{2}+\frac{77}{2}i}

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