# Struggling to remember something small

Evaluate $$(1+i)^{11}$$.

I got $$(1+i)^{11} = 2^{11/2} [\cos (11/4π) + i\sin(11/4π)]$$

I don't know how the angle change to $$3/4\pi$$

That is:
$$2^{11/2} [\cos(3/4π) + i\sin(3/4π)]$$

I think I have forgotten

• Hint: $\frac{11}{4}\pi=2 \pi + \frac{3}{4}\pi$ – dxiv Dec 9 '16 at 17:17

$$(1+i)^{11}=(1+i)^{10}\times(1+i)$$ $$(2i)^5\times(1+i)=32i(1+i)=32(-1+i)$$
$\cos(x)=\cos(x + 2\pi n)$ where $n\in\mathbb{Z}$, it is a periodic function, as is $\sin$. So a complex number with an argument of $11\pi/4$ is just the same as one with argument $3\pi/4$, you can change it by integer multiples of $2\pi$. The complex number lies in the second quadrant (top left)