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Power series for the sine
The sine can be expressed as a power series
Explanation
As the sine can be differentiated continuously, you get
The point x = 0 gives sin (0) = 0 and cos (0) = 1 so you get
We substitute this in the Taylor series and find
so
Symmetry
The power series contains only odd numbers, so there is the symmetry
Example 1
You can see that sin (0) = 0, because
sin (0) = 0 − 0 + 0 − ... = 0