Find z1z2 and z1z2 Write the answer in the standard form z

Find z_1z_2 and z_1z_2. Write the answer in the standard form z = a + bi. z_1 = 10(cos 7 pi/6 + i sin 7 pi/6, z_2 = 1/5(cos pi/6 + i sin pi/6) z_1z_2 = z_1/z_2 =

Solution

z1 = 10(cos7pi/6 +isin7pi/6) = 10e^i*7pi/6

z2 = (1/5)(cospi/6 + i*sin pi/6) = (1/5)e^i*pi/6

z1*z2 = 10*1/5e^(i*8pi/6)

= 2e^i*4pi/3)

= 2[cos4pi/3 + isin4pi/3]

= 2[-1/2 - i*sqrt3/2 ] = -1 - i*sqrt3

z1/z2 = 10e^i*7pi/6/(1/5)e^i*pi/6

=50e^i*(pi/6)

= 50[cospi/6 +isinpi/6]

= 25sqrt3+i*25