Let A = pi/12 and B = 5pi/12
A + B = pi/12 + 5pi/12 = 6pi/12 = pi/2
By replacing the values with A and B, we have
w1 = 4(CosA + iSinA)
w2 = 3(CosB + iSinB)
To multiply w1 and w2, we would multiply each term in the first parentheses by each term in the second one. We have
w1 * w2 = 4(CosA + iSinA) * 3(CosB + iSinB)
= 4 * 3(CosA + iSinA) * (CosB + iSinB)
= 12[CosACosB + iCosASinB + iSinACosB +i^2SinASinB]
Recall, i^2 = - 1
Thus, we have
12[CosACosB + iCosASinB + iSinACosB - SinASinB]
By rearranging the terms inside the parentheses, we have
12[CosACosB - SinASinB + iSinACosB + iCosASinB]
12[CosACosB - SinASinB + i(SinACosB + CosASinB)]
Recall the following trigonometric identities
CosACosB - SinASinB = Cos(A + B)
SinACosB + CosASinB = Sin(A + B)
By applying these identities, we have
12[Cos(A + B) + iSin(A + B)]
Replacing A + B with pi/2, we have
12[Cos(pi/2) + iSin(pi/2)] expression 6
Recall, pi = 180 degrees
pi/2 = 180/2 = 90 degrees
Also,
Cos90 = 0
Sin90 = 1
By putting these values into expression 6, it becomes
12[Cos90 + iSin90]
12(0 + i)
= 12i
Option B is correct