The Derivatives of the Sine and Cosine. [from Sensible Calculus]
(i) The Derivative of the Sine at $x=0$: $\sin'(0)=1$.
Proof:
Check this visually with the mapping diagram CDD.DSMD0
CDD.DSMD0
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(ii) The Derivative of the Cosine at $x=0$: $\cos'(0)=0$.
Proof:
Check this visually with the mapping diagram CDD.DCMD0
CDD.DCMD0
Download GeoGebra file
Now that we have found the derivatives for
sine and cosine at $ x =0$,
it is time to think of these derivatives more generally. In Theorem CCD.DSC we state these results.
Details for the proofs can be found in most
calculus texts or in The Sensible Calculus Book Section I.F.3 Derivatives of the sine and cosine functions.
What is missing in other proofs are visualizations using mapping
diagrams. These are provided with the mapping diagram CCD.DSCMD.
Theorem CCD.DSC. i) $D_x\sin(x) = \sin'(x) =\cos(x)$.
ii) $D_x\cos(x) = \cos'(x) = -\sin(x)$
Use the slider "Choose s(x):" to switch between the visulization for $s(x)=\sin(x)$ and $s(x)=\cos(x)$.
The Steps refer to the four steps for finding a derivative in CCD.DDN4S.