There are plenty of connections between trigonometry and the complex plane. One particular comparison seems striking to me, and that’s the four-step cycle inherent in exponentiating $i$ to various powers and taking derivatives of $\sin$. Namely:
$$i^0=1, i^1=i, i^2=-1, i^3=-i, \dots$$ $$f(x)=\sin x,f’(x)=\cos x,f’’(x)=-\sin x,f’’’(x)=-\cos x,\dots$$
I’m aware of the connection between $e^{i\theta}=i\sin\theta+\cos\theta$ and $\frac{d\left(e^{it}\right)}{dt}=ie^{it}$, so there’s already a connection between trigonometry, derivatives, and the unit circle in the imaginary plane. Is there an extension of this, or something else entirely, that connects the four-step cycle of differentiating $\sin$ and that of exponentiating $i$?
H/t Mark S for pointing out the obvious.
Taking the equation $e^{ix}=i\sin x+\cos x$ and differentiating it four times yields the following:
$$e^{ix}=i\sin x+\cos x\\ie^{ix}=i\cos x-\sin x\\-e^{ix}=-i\sin x-\cos x\\-ie^{ix}=-i\cos x+\sin x\\\dots$$
The derivative of the left side rotates through the powers of $i$ and the derivative of the right side rotates through the derivatives of $\sin$ and $\cos$.
Excuse me while I go repeatedly smack myself in the face.