Question

Let z1 = 4(cos 135° + i sin 135°) and z2 = 2(cos 45° + i sin 45°). Write the rectangular form of z1z2.

Can someone help me ? Explain

Answer 1

135 degrees = 3pi / 4
45 degrees = pi/4

cos(t) + i * sin(t) = e^(t * i)

z1 = 4 * e^((3pi/4) * i)
z2 = 2 * e^((pi/4) * i)

Now we just use the rules of exponents and expand

z1 * z2 =>
4 * 2 * e^((3pi/4) * i + (pi/4) * i) =>
8 * e^(pi * i) =>
8 * (cos(pi) + i * sin(pi)) =>
8 * (-1 + i * 0) =>
8 * (-1 + 0) =>
8 * (-1) =>
-8

Comments

z1/z2=4 * e^((3pi/4) * i)/2 * e^((pi/4) * i)
         =2*e^((3pi/4-pi/4)i
         =2e(pi/2)i
         =2(cospi/2+isinpi/2)
         =2i