Mnemonic for derivative/integral of $\sin x$ and $\cos x$

Question

I'd love to know if anyone has a good mnemonic for answers of the following:

$$\frac{\mathrm{d}}{\mathrm{d}x} \, \sin x$$

$$\frac{\mathrm{d}}{\mathrm{d}x} \, \cos x$$

$$\int \sin x \,\mathrm{d}x$$

$$\int \cos x \,\mathrm{d}x$$

I know the first one by heart, and derive the others from it. This sometimes takes me up to five seconds, longer if I'm not really thinking clearly! Does anyone have a good mnemonic for the answers to these common occurrences?

Answer 1

In the trigonometric unit circle. Differentiation is clockwise and integration is anticlockwise. For example if you were to search for the derivative on $\sin(x)$ then clockwise next is the derivative which is $\cos(x)$. Similarly, the antiderivative of $-\sin(x)$ is found by going anticlockwise on the unit circle, and we get $\cos(x)$.

source: https://getrevising.co.uk/https_proxy/1364

Answer 2

Vulgar, but how I remember it:

$$ \begin{matrix} D & I \\ C & S \end{matrix}~~~~~~~~\text{ are negative } $$

Derivative/integral of cosine and sine are negative.

Answer 3

$$\large \sin(x) \xrightarrow{\huge \frac{\text{d}}{\text{d}x}} \cos(x) \xrightarrow{\huge \frac{\text{d}}{\text{d}x}} -\sin(x) \xrightarrow{\huge \frac{\text{d}}{\text{d}x}} -\cos(x) \xrightarrow{\huge \frac{\text{d}}{\text{d}x}} \sin(x)$$

$$\\\\$$

$$\large \sin(x) \xrightarrow{\huge \int} - \cos(x) \xrightarrow{\huge \int} -\sin(x) \xrightarrow{\huge \int} \cos(x) \xrightarrow{\huge \int} \sin(x)$$

$$\\\\$$

(leaving apart the constants of integration).