1. Given z1 = 2(cos pi/6 + i sin pi/6) and z2 = 3 (cos pi/4 + i sin pi/4), find z1z2 where 0 ≤ theta ≤ 2pi
2. Find the product of z1 = 2/3 (cos60° + i sin60°) and z2 = 9 (cos20° + i sin20°) where 0 ≤ theta ≤ 360°
3. Given z1 = 12 (cos pi/3 + i sin pi/3) and z2 = 3 (cos 5pi/6 + i sin 5pi/6), find z1/z2 where 0 ≤ theta ≤ 2pi
4. Find the quotient of z1 = cos 2pi/3 + i sin 2pi/3 and z2 = 2 (cos pi/12 + i sin pi/12) where 0 ≤ theta ≤ 2pi
1. To find the product z1z2, we multiply their magnitudes and add their angles:
z1z2 = (2)(3)[cos(pi/6 + pi/4) + i sin(pi/6 + pi/4)] = 6 [cos(5pi/12) + i sin(5pi/12)]
So, z1z2 = 6(cos 5pi/12 + i sin 5pi/12).
2. To find the product z1z2, we multiply their magnitudes and add their angles:
z1z2 = (2/3)(9)[cos(60° + 20°) + i sin(60° + 20°)] = 6[cos(80°) + i sin(80°)]
So, z1z2 = 6(cos 80° + i sin 80°).
3. To find the quotient z1/z2, we divide their magnitudes and subtract their angles:
z1/z2 = (12/3)[cos(pi/3 - 5pi/6) + i sin(pi/3 - 5pi/6)] = 4[cos(-pi/2) + i sin(-pi/2)]
So, z1/z2 = 4(cos -pi/2 + i sin -pi/2).
4. To find the quotient z1/z2, we divide their magnitudes and subtract their angles:
z1/z2 = (1/2)[cos(2pi/3 - pi/12) + i sin(2pi/3 - pi/12)] = (1/2)[cos(7pi/12) + i sin(7pi/12)]
So, z1/z2 = (1/2)(cos 7pi/12 + i sin 7pi/12).