Find the imaginary part of\[(\cos12^\circ+i\sin12^\circ+\co s48^\circ+i\sin48^\circ)^6.\]

The imaginary part is 0

Step-by-step explanation:

The number given is:

First, we can expand this power using the binomial theorem:

After that, we can apply De Moivre's theorem to expand each summand:

The final step is to find the common factor of i in the last expansion. Now:

The last part is to multiply these factors and extract the imaginary part. This computation gives:

(It is not necessary to do a lengthy computation: the summands of the imaginary part are the products sin(a)cos(b) and cos(a)sin(b) as they involve exactly one i factor)

A calculator simplifies the imaginary part Im(x⁶) to 0