For z1 = 4cis (7pi/6) and z2 = 3cis(pi/3), find z1 * z2 in rectangular form.

Answer 1

short cut
4cis (7pi/6)*3cis(pi/3)
= 12 cis(7π/6+π/3) , by De Moivre's theorem
= 12cis(3π/2)
but you wanted it in rectangular form
= 12(cos 3π/2 + i sin 3π/2)
= 12(0 - 1 i)
= -12i

long way (if you don't know the above law):
4cis (7pi/6)
= 4(cos 7π/6 + i sin 7π/6)
= 4(-√3/2 - 1/2 i)
= (-2√3 - 2i)

3cis(pi/3)
= 3(1/2 + √3/2 i)
= (3/2 + 3√3/2 i)

4cis (7pi/6)*3cis(pi/3)
= (-2√3 - 2i)(3/2 + 3√3/2 i)
= -3√3 - 9i - 3i - 6√3 i^2
= -12 i