Question

Find the product of the complex numbers. Leave answer in polar form.z1 = 5(cos 20° + i sin 20°)z2 = 4(cos 10° + i sin 10°)Group of answer choices9(-cos 200° - i sin 200°)20(cos 200° + i sin 200°)9(cos 30° + i sin 30°)20(cos 30° + i sin 30°)

Answer 1

We need to find the product of the given complex numbers.

Knowing that i² = -1, we obtain:

`\begin{gathered} z_1\cdot z_2=5\mleft(\cos 20\degree+i\sin 20\degree\mright)\cdot4\mleft(\cos 10\degree+i\sin 10\degree\mright) \\ \\ =20(\cos 20\degree\cos 10\degree+i^2\sin 20\degree\sin 10\degree+i\sin 20\degree\cos 10\degree+i\sin 10\degree\cos 20\degree) \\ \\ =20\lbrack(\cos 20\degree\cos 10\degree-\sin 20\degree\sin 10\degree)+i(\sin 20\degree\cos 10\degree+\sin 10\degree\cos 20\degree)\rbrack \\ \\ =20\lbrack\cos (20\degree+10\degree)+i\sin (20\degree+10\degree)\rbrack \\ \\ =20(\cos 30\degree+i\sin 30\degree) \end{gathered}`