Calculate (1 − i)(− \sqrt{3} + i) and \frac{2 + 2i}{1 + \sqrt{3}i } using polar form.
Question 9.1.11: Calculate (1 − i)(− √3 + i) and 2 + 2i / 1 + √3i using polar...
(1 − i)(− \sqrt{3} + i) = \sqrt{2} \left(\cos (- \frac{\pi }{4} ) + i \sin (- \frac{\pi }{4} ) \right) 2 \left(\cos ( \frac{5\pi }{6} ) + i \sin ( \frac{5 \pi }{6} )\right)
= 2 \sqrt{2} \left(\cos (- \frac{\pi }{4} + \frac{5 \pi }{6} ) + i \sin (- \frac{\pi }{4} + \frac{5 \pi }{6}) \right)
= 2 \sqrt{2} \left(\cos ( \frac{7 \pi }{12} ) + i \sin ( \frac{7 \pi }{12} ) \right)
\frac{2 + 2i}{1 + \sqrt{3} i } = \frac{ 2 \sqrt{2} \left(\cos ( \frac{\pi }{4} ) + i \sin (\frac{\pi }{4} ) \right)}{2 \left(\cos (\frac{\pi }{3} ) + i \sin (\frac{\pi }{3} ) \right)}
= \sqrt{2} \left(\cos ( \frac{\pi }{4} – \frac{\pi }{3} ) + i \sin (\frac{\pi }{4} – \frac{\pi }{3}) \right)
= \sqrt{2} \left(\cos (- \frac{\pi }{12} ) + i \sin (- \frac{\pi }{12} ) \right)
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