Question

[2(cos15° + isin15°)]^4

Answer 1

Answer:

$8+8\sqrt{3} \ i$

Step-by-step explanation:

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

De Moivre's Formula :

$(\cos x+i\sin x)^{n}=\cos n x+i\sin nx\\\\\\(\cos 15+i \sin15)^{4}=\cos(4\times 15)+i\sin (4\times 15)\\=\cos60+i\sin60\\=\frac{1}{2} +\frac{\sqrt{3}}{2} \\=\frac{1+\sqrt{3i} }{2}$

Now from eqn(1)

$[2(\cos15+i\sin15)]^{4}=16\times \frac{1+\sqrt{3}i}{2} =8+8\sqrt{3}i$