Find z1 z2 and z1z2 for z1 6 cos 162 degree i sin 162 degr

Find z_1 z_2 and z_1/z_2 for z_1 = 6 (cos 162 degree + i sin 162 degree), z_2 = 5 (cos 72 degree + i sin 72 degree). Write each answer in polar form. z_1 z_2 = [cos degree + i sin degree] (Simplify your answers. Typo any angle measures in degrees. Use angle measures greater than or equal to 0 and less than 360.) z_1/z_2 = [cos degree + i sin degree] (Simplify your answers. Type any angle measures in degrees. Use angle measures greater than or equal to 0 and less than 360.)

Solution

We can use:
(a) [r(cos + i sin )][r2(cos 2 + i sin )] = r1r2[cos( + ) + i*sin( + )]
(b) [r(cos + i sin )]/[r2(cos 2 + i sin )] = (r/r)[cos( + ) - i*sin( + )].

Thus:
zz
= 6*5*(cos 162 + i sin 162)(cos 72 + i sin 72)
= 30*cos(162 + 72) + i*sin(162 + 72), from (a)
= 30*[cos 234 + i sin 234]

and:
z/z
= (6/5)*(cos 162 + i sin 162)/(cos72 + i sin 72)
= (6/5)*[cos(162 - 72) + i*sin(162 - 72)], from (b)
= (6/5)*[cos(90) + i*sin(90)] = (6/5)*i

I hope this helps!