From the information given,
w1 = 5(cos(15) + isin(15))
w2 = 3(cos(45) + isin(45))
Let A represent 15 degrees and B represent 45 degrees
To find w1 * w2, we would multiply each term in the first parentheses by each term in the second parentheses. We have
w1 * w2 = 5(Cos(A) + iSin(A)) * 3(Cos(B) + iSin(B))]
= 15[CosACosB + CosA * iSinB + iSinACosB + i^2SinASinB]
= 15[CosACosB + i(CosASinB + SinACosB) + i^2SinASinB]
= 15[CosACosB + i(SinACosB + CosASinB) + i^2SinASinB]
Recall, i^2 = - 1
Thus, we have
15[CosACosB + i(CosASinB + SinACosB) - SinASinB]
15[CosACosB - SinASinB + i(SinACosB + CosASinB)]
Recall the following trigonometric identities
Cos(A + B) = CosACosB - SinASinB
Sin(A + B) = SinACosB + CosASinB
By applying these identities, we have
15[Cos(A + B) + i(Sin(A + B)]
By replacing A with 15 and B with 45, we have
15[Cos(15 + 45) + i(Sin(15 + 45)]
= 15[(Cos60 + iSin60]
Option 1 is correct