Simplify expression $e^{i\pi/3}(\sqrt{2}e^{i\pi /4} -1)$

Question

Problem

Simplify expression $e^{i\pi/3}(\sqrt{2}e^{i\pi /4} -1)$

Attempt to solve

$$= \sqrt{2}e^{i 7\pi/12}-e^{i \pi / 3} = e^{i\frac{5}{6}\pi}$$

I can't seem to find how this expression is $e^{i\frac{5}{6}\pi}$ ?

I would like the final result be in polar form $re^{i\theta}$

Answer 1

It might be slightly simpler to notice that $$ \sqrt{2}e^{i\pi/4}-1 = \sqrt{2}\left(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\right)-1=i=e^{i\pi/2}, $$ so $$ e^{i\pi/3}(\sqrt{2}e^{i\pi/4}-1)=e^{i\pi/3}e^{i\pi/2}=e^{i5\pi/6}. $$

Answer 2

One may write trigonometric form \begin{align} e^{i\frac{pi}{3}}(\sqrt{2}e^{i\frac{\pi}{4}} -1) &= (\cos\dfrac{\pi}{3}+i\sin\dfrac{\pi}{3})\left(\sqrt{2}(\cos\dfrac{\pi}{4}+i\sin\dfrac{\pi}{4})-1\right) \\ &= \left(\dfrac{1}{2}+i\dfrac{\sqrt{3}}{2}\right)\left(1+i-1\right) \\ &= \dfrac{-\sqrt{3}}{2}+i\dfrac{1}{2} \\ &= e^{\frac56\pi i} \end{align}