Turning Round And Round

And it seem that,

$a^{\frac{1}{m}}+b^{\frac{1}{n}}=c^\frac{1}{k}$

is just,

$cos(\theta_m)+cos(\theta_n)=cos(\theta_k)$, where the radius of the circle has been normalized with a factor $\cfrac{\pi}{2}$.

$r\theta_n=arc_n$ , $r=\cfrac{2}{\pi}$

$\theta_n=\cfrac{1}{n}\cfrac{\pi}{2}$

$\theta_m=\cfrac{1}{m}\cfrac{\pi}{2}$

$\theta_k=\cfrac{1}{k}\cfrac{\pi}{2}$

$cos(\cfrac{\pi}{2m})+cos(\cfrac{\pi}{2n})=cos(\cfrac{\pi}{2k})$

What happened to $a^{m}, b^{n}, c^{k}$?

$${ln(y)nln(a)}=\frac{\pi}{2}ln(a)$$   

Consider, $$y=e^{i\theta}$$

$$ln(y)=i\cfrac{1}{n}\cfrac{\pi}{2}$$

$$\cfrac{2ln(y)ln(a)}{\pi}=i\frac{1}{n}ln(a)$$

$$ln(y)nln(a)=i\frac{\pi}{2}ln(a)$$

$$ln(y)=ln(a)(\frac{i\pi}{2}-n)$$

$$e^{i\theta}=a^{\cfrac{i\pi}{2}-n}$$

$$a^n=e^{-i\theta} a^{\cfrac{i\pi}{2}}$$

And so,

$$a^m=e^{-i\theta_m} a^{\cfrac{i\pi}{2}}$$

$$b^n=e^{-i\theta_n} a^{\cfrac{i\pi}{2}}$$

$$c^k=e^{-i\theta_k} a^{\cfrac{i\pi}{2}}$$

With $\theta_n=\cfrac{\pi}{2n}$,  $\theta_m=\cfrac{\pi}{2m}$,  $\theta_k=\cfrac{\pi}{2k}$

$$a^m+b^n=e^{-i\cfrac{\pi}{2n}}. a^{i\cfrac{\pi}{2}}+e^{-i\cfrac{\pi}{2m}} .b^{i\cfrac{\pi}{2}}=e^{-i\cfrac{\pi}{2k}}. c^{i\cfrac{\pi}{2}}=c^k$$

And they just keep turning.