Question

Find the product of z1 and z2, where z1 = 2(cos 80° + i sin 80°) and z2 = 9(cos 110° + i sin 110°)

Answer 1

When multiplying complex numbers, you multiply the moduli and add the arguments:

$z_1z_2=2\times9(\cos(80^\circ+110^\circ)+i\sin(80^\circ+110^\circ))=18(\cos190^\circ+i\sin190^\circ)$

Answer 2

ANSWER

$z_1 z_2=18( \cos( 190 \degree ) + i \sin( 190 \degree) )$


EXPLANATION

The given complex numbers are;

$z_1 = 2( \cos(80 \degree) + i \sin(80 \degree) )$

and


$z_2= 9( \cos(110 \degree) + i \sin(110 \degree) )$


Recall the binominal identity for complex numbers in polar form.



$z_1 z_2=r_1 r_2( \cos( \theta_1 +\theta_2) + i \sin(\theta_1 +\theta_2) )$



We apply this property to obtain,


$z_1 z_2=2 \times 9( \cos( 80 \degree + 110 \degree ) + i \sin(80 \degree + 110 \degree) )$



$z_1 z_2=18( \cos( 190 \degree ) + i \sin( 190 \degree) )$