Question

[2(cos15° + isin15°)]^4

16 + 16 square rt 3 i
8 + 8square rt 3 I
16 square rt 3 + 16i
8 square rt 3 + 8i

Answer 1

`8+8√(3) \ i`

Explanation:

`[2(\cos15+i15)^(4)]\\=2^(4)(\cos15+i\sin15)^(4)\\=16(\cos15+i\sin15)^(4)..........(1)`

De Moivre's Formula :

`(\cos x+i\sin x)^(n)=\cos n x+i\sin nx\\\\\\(\cos 15+i \sin15)^(4)=\cos(4* 15)+i\sin (4* 15)\\=\cos60+i\sin60\\=(1)/(2) +(√(3))/(2) \\=(1+√(3i) )/(2)`

Now from eqn(1)

`[2(\cos15+i\sin15)]^(4)=16* (1+√(3)i)/(2) =8+8√(3)i`