# Deduce that $\tan \frac{3\pi}{8}=1+\sqrt2$

Find the modulus and the argument, in radians in terms of $\pi$, of

$$z_1=\frac{1+i}{1-i}, z_2=\frac{\sqrt2}{1-i}, z_3=\left(\frac{1+i}{1-i}\right)^2$$

Plot $z_1, z_2$ and $z_1+z_2$ on an Argand diagram.

Deduce that $\tan \dfrac{3\pi}{8}=1+\sqrt2$

I found $|z_1|=|z_2|=|z_3|=1$ and $\arg z_1=\frac{\pi}{2}, \arg z_2=\frac{\pi}{4},\arg z_3=\pi$. I plotted the Argand diagram. I cannot see how this leads on to the final part of the question. Thanks in advance.

• Hint: what is $\arg(z_1+z_2)$? – Macavity Aug 28 '13 at 19:51

$$\tan{\frac{3 \pi}{8}} = \frac{\sin{(3 \pi/8)}}{\cos{(3 \pi/8)}} = \sqrt{\frac{1-\cos{(3 \pi/4)}}{1+\cos{(3 \pi/4)}}} = \sqrt{\frac{2+\sqrt{2}}{2-\sqrt{2}}} = \frac{2+\sqrt{2}}{\sqrt{2}} = \sqrt{2}+1$$

• Been trying to make this more relevant to the actual problem from the OP, but the editing tool has been crashing my browser. – Ron Gordon Aug 28 '13 at 20:21

Hint: Find $z_1 + z_2$ and use $$\tan(\operatorname{arg}z) = \frac{\operatorname{Im} z}{\operatorname{Re} z}.$$

Use tip suggested by @njguliyev