, where 
is the nth derivative of f. Since this approximates the sine function best when x is close to c, we take 
radians.
Taylor Series:
.
We need to expand sin(x) with a Taylor series: , where is the nth derivative of f. Since this approximates the sine function best when x is close to c, we take radians.
For convenience we create the following table with c set to 
:
|
n |
|
|
|
0 |
|
|
|
1 |
|
|
|
2 |
|
|
|
3 |
|
|
Taking the successive derivatives, we can fill in the middle column:
|
n |
|
|
|
0 |
|
|
|
1 |
|
|
|
2 |
|
|
|
3 |
|
|
Now we evaluate the derivatives at 
|
n |
|
|
|
0 |
|
1 |
|
1 |
|
0 |
|
2 |
|
-1 |
|
3 |
|
0 |
Substitute the values from the table into the first 4 terms of the Taylor series:
,
which simplifies to 
.
Since 
radians, this becomes 
Evaluating this, we get 
