Evaluate $\left(1+\sin\frac{\pi}{5}+i\cos\frac{\pi}{5}\right)^5+i\left(1+\sin\frac{\pi}{5}-i\cos\frac{\pi}{5}\right)^5$

Evaluate $$\left(1+\sin\frac{\pi}{5}+i\cos\frac{\pi}{5}\right)^5+i\left(1+\sin\frac{\pi}{5}-i\cos\frac{\pi}{5}\right)^5$$

I did this by $$\left(1+\sin\frac{\pi}{5}+i\cos\frac{\pi}{5}\right)^5=\left(1+\cos\frac{3\pi}{10}+i\sin\frac{3\pi}{10}\right)^5$$ and get $$0$$

Does anyone have another idea?

Thanks

• Did you use the complex e-power? – imranfat Oct 29 '19 at 13:00
• "Does anyone have another idea?" Could you show us your method? – The Jade Reaper Oct 29 '19 at 13:01
• math.stackexchange.com/questions/936196/… this post maybe help you! – Riemann Oct 29 '19 at 13:05
• @The Jade Reaper, I did this by $\left(1+\sin\frac{\pi}{5}+i\cos\frac{\pi}{5}\right)^5=\left(1+\cos\frac{3\pi}{10}+i\sin\frac{3\pi}{10}\right)^5$ – user62498 Oct 29 '19 at 13:06

Say $$z = \sin\frac{\pi}{5}+i\cos\frac{\pi}{5}$$.

$$\big($$We have $$-iz = \cos\frac{\pi}{5}-i\sin\frac{\pi}{5} = \cos\frac{-\pi}{5}+i\sin\frac{-\pi}{5}$$, so $$\boxed{z^5 = -i}$$ by De'Moivre formula.$$\big)$$

Notice that $$\bar{z}={1\over z}$$.

Then your expression is $$\begin{eqnarray}w &=& (1+z)^5+i(1+\bar{z})^5\\ &=& (1+z)^5+i(1+{1\over z})^5\\ &=& (1+z)^5+i{(z+1)^5\over z^5}\\ &=& (1+z)^5+i{(z+1)^5\over -i}\\ &=& 0\\ \end{eqnarray}$$

• Aqua,I know this way but I don't use binomial theorem – user62498 Oct 29 '19 at 13:09
• @ Aqua, this is great! thanks – user62498 Oct 29 '19 at 13:25

$$\frac{1+\sin x+i\cos x}{1+\sin x-i\cos x}=\frac{(1+\sin x+i\cos x)^2}{(1+\sin x)^2+\cos^2 x}$$

$$=\frac{2(1+\sin x)(\sin x+i\cos x)}{2(1+\sin x)}=\cos\left(\frac\pi2-x\right)+i\sin\left(\frac\pi2-x\right).$$ Apply de Movire's Theorem, you can get $$\left(\frac{1+\sin x+i\cos x}{1+\sin x-i\cos x}\right)^n=\cos n\left(\frac{\pi }{2}-x\right)+i\sin n\left(\frac{\pi}{2}-x\right).$$ In youe situation, take $$x=\pi/5$$. $$\left(1+\sin\frac{\pi}{5}+i\cos\frac{\pi}{5}\right)^5+i\left(1+\sin\frac{\pi}{5}-i\cos\frac{\pi}{5}\right)^5$$ $$=\left(\frac{1+\sin\frac{\pi}{5}+i\cos\frac{\pi}{5}}{1+\sin\frac{\pi}{5}-i\cos\frac{\pi}{5}}\right)^5+i=(-i)^5+i=0.$$

Using the facts that $$(e^{\pi i/5})^5=e^{\pi i}=-1$$ and $$i^5=i$$, we have

\begin{align} (1+\sin\pi/5+i\cos\pi/5)^5+i(1+\sin\pi/5-i\cos\pi/5) &=(1+ie^{-\pi i/5})^5+i(1-ie^{\pi i/5})^5\\ &=-(e^{\pi i/5})^5(1+ie^{-\pi i/5})^5+i^5(1-ie^{\pi i/5})^5\\ &=-(e^{\pi i/5}+i)^5+(i+e^{\pi i/5})^5\\ &=0 \end{align}