Maclaurin series of the sine function
Categories: maclaurin series
In this section we will use the Maclaurin series to find a polynomial approximation to the sine function, sin(x).
The general formula for the Maclaurin series for the function f(x) is:
Where:
- f(0) is the value of the function for x = 0.
- f'(0) is the value of the first derivative function for x = 0.
- f''(0) is the value of the second derivative function for x = 0.
- f'''(0) is the value of the third derivative function for x = 0.
- And so on.
To apply this to the sine function, we need to calculate those derivatives.
Derivatives the sine function
In our case f(x) is the sine function:
The first derivative of the sine function is the cosine function:
We find the second derivative by differentiating again. If we differentiate the cosine function we get negative sine:
We find the third derivative by differentiating again. If we differentiate negative sine function we get negative cosine:
Finally, we find the fourth derivative by differentiating yet again. If we differentiate negative cosine function we get sine:
We are back to the sine function again. If we differentiate again, the pattern will repeat: s, c, -s, -c, s, c, -s, -s ...
Values of the derivatives at x = 0
The sine of 0 is 0, and the cosine of 0 is 1. This allows us to calculate the following values for f(x) and its derivatives when x = 0: This means that:
- f(0) = 0
- f'(0) = 1
- f''(0) = 0
- f'''(0) = -1
Again, this patten repeats for higher order derivatives: 0, 1, 0, -1, 0, 1, 0, -1 ...
Maclaurin series of sine function
Taking the general equation above:
We can replace f(0), f'(0), and all of the higher order derivatives with the values we found above:
This can be tidied up by removing the zero terms. We have also added some extra terms up to teh term in the 7th power of x:
Or using sigma notation (as described here):
A graphical illustration of the Maclaurin expansion
Here is an animation that shows the first 4 non-zero terms of the expansion being added in one by one:
One way to gain an intuitive insight into how the Maclaurin expansion works is to look at graphs of the approximation as we add the terms one by one.
Step 1
Taking just the first term of the expansion gives us:
This graph shows the expansion (in yellow) and the sine function (in black):
This approximation is not very good, it is just a diagonal straight line. It has the same value and slope at the sine function at 0, but it diverges away from that point.
Step 2
Taking the first two terms of the expansion gives us:
Here is the graph:
This is a better approximation, and will continue to improve as we add more terms.
Step 3
Taking the first three terms of the expansion gives us:
Here is the graph:
Step 4
Taking the first four terms of the expansion gives us:
Here is the graph:
See also
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