Generalized Trigonometric Functions in terms of exponentials and roots of unity

Question

I am trying to come up with generalized trigonometric functions using the exponential definition that we use today for the trig functions sine and cosine $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}; \cos x =\frac{e^{ix}+e^{-ix}}{2}$$ I was then thinking, in terms of roots of unity, if we define $$\omega_n^k=e^\frac{2ki\pi}{n}$$ Then $$\cos x =\frac{(\omega_4^0)^0e^{\omega_4^1x}+(\omega_4^2)^0e^{\omega_4^3x}}{2\omega_4^0};\sin x =\frac{(\omega_4^0)^1e^{\omega_4^1x}+(\omega_4^2)^1e^{\omega_4^3x}}{2\omega_4^1}$$ Then I could extend as say $$C_0(x)=\frac{(\omega_6^0)^0e^{\omega_6^1x}+(\omega_6^2)^0e^{\omega_6^3x}+(\omega_6^4)^0e^{\omega_6^5x}}{3\omega_6^0}$$ $$C_1(x)=\frac{(\omega_6^0)^1e^{\omega_6^1x}+(\omega_6^2)^1e^{\omega_6^3x}+(\omega_6^4)^1e^{\omega_6^5x}}{3\omega_6^1}$$ $$C_2(x)=\frac{(\omega_6^0)^2e^{\omega_6^1x}+(\omega_6^2)^2e^{\omega_6^3x}+(\omega_6^4)^2e^{\omega_6^5x}}{3\omega_6^2}$$ I've seen generalizations for trig function using the integral definition of the inverses, and i've seen the defined using lacunary power series. WOuld this be another way to define a trig generalization?