z1I Find xw and zw for each pair of complex numbers First co

Find xw and z/w for each pair of complex numbers. First convert to polar, then perform the product and quotient is polar. Leave your answer is polar form. z = 2 + 2i, w = squareroot 3 - i z = 1 + i, w = squareroot 3 - i Write the expression in the standard form a + bi. Use De Moivre\'s Theorem.

Solution

z = 1+i = 1*e^(ipi/4)

, w = sqrt3 - i = 2e^(-ipi/6)

z/w = (1+i)/[sqrt(3) -i] = e^(ipi/4)/2e^(-ipi/6)

= 0.5e^i*(10/24) = 0.5e^i(5/12)

= 0.5[cos5pi/12 +i*sin5pi/12]

= 0.13 + i*0.48

z*w = (1+i)*[sqrt(3) -i]

= 2e^(i*pi/4) *e^(-i*pi/6)

= 2e^(i*pi/12)

= 2[cospi/12 + i*sinpi/12]

= 1.93 + i*0.52