Express $z=4\sqrt2(1+i)$ in modulus/argument form. Hence find the two square roots of $z$ and mark their representations on an Argand Diagram.
So far I've worked out the mod/arg form of the complex number which is just
$$ z = 8(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}) \\ $$
Then used $\alpha = \frac{\theta +360k}{n}$ where $k$ is $0,1,2,...,n-1$ and got two results for the argument. The first being $\frac{-7\pi}{8}$ and the other being $\frac{\pi}{8}$.
My final answers were
$$ z_1 = 2\sqrt2(\cos\frac{-7\pi}{8} + i\sin\frac{-7\pi}{8}) \\ z_2 = 2\sqrt2(\cos\frac{\pi}{8} + i\sin\frac{\pi}{8}) $$
For some reason, the answers from the book were
$$ z_1 = 2\sqrt2(\cos\frac{\pi}{8} + i\sin\frac{\pi}{8}) \\ z_2 = -2\sqrt2(\cos\frac{\pi}{8} + i\sin\frac{\pi}{8}) $$
What was it that I did wrong in my calculations? Thanks in advance!
Both are correct. In fact, they are the same.
$$ 2\sqrt 2 (\cos\frac{-7\pi}{8} + i\sin\frac{-7\pi}{8}) = -2 \sqrt 2 (\cos\frac{\pi}{8} + i\sin\frac{\pi}{8}) $$
Moreover, $-2\sqrt 2 (\cos\frac{\pi}{8} + i\sin\frac{\pi}{8})$ looks better.