Answer to Question #203412 in Complex Analysis for Rajay Myrie

Question

Given that 𝑧1 = 3 + 𝑖 𝑎𝑛𝑑 𝑧2 = 2 − 𝑖: i. Find the modulus and argument of 𝑧1 𝑧2 (5 marks) ii. Express 𝑧1 𝑧2 in polar and exponential form iii. Use de Moivre’s theorem to find an expression for ( 𝑧1 𝑧2 ) 4 

Answer 1

$\displaystyle i.)\\ z_1 = 3 + i,\,\, z_2 = 2 - i\\ z_1 z_2 = (3 + i)(2 - i) = 6 - 3i + 2i + 1 = 7 - i\\ |z_1 z_2| = \sqrt{7^2 + (-1)^2} = \sqrt{50} = 5\sqrt{2}\\ \arg(|z_1 z_2|) = \arctan\left(\frac{-1}{7}\right) = \frac{3\pi}{2} + \frac{\pi}{2} - 0.142 = 6.141\\ ii.)\\ z_1 z_2 = \sqrt{50}e^{6.141i}\\ z_1 z_2 = \sqrt{50}(\cos(6.141) + i\sin(6.141))\\ iii. \\ (z_1 z_2)^4 = 2500(\cos(6.141\times 4) + i\sin(6.141\times 4)) = 2500(\cos(25.56) + i\sin(25.56))$