Finding the Maclaurin Series for Cos(x)

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  • 0:04 Taylor Series
  • 3:53 Approximating the Cosine
  • 5:39 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we show how to find the Maclaurin series for cos(x). We'll also work through the details of how to apply the Maclaurin series for cos(x) when approximating the cosine of an arbitrary value.

Taylor Series

You may have used the Taylor series in some of your courses. In this lesson, we'll explore a series based on the Taylor series.

Our cosine series example will give an answer you could easily obtain with your calculator. So why bother with the series? Well, what if we had some value of x whose cosine was not known? Or what if we wanted to prove the derivative of sin(x) is cos(x)? This series is the math skill for just such applications.

Basically, if we have a function we can differentiate, the Taylor series provides a sum of the terms which will approximate the function. The compact way to write the Taylor series uses summation notation:


Taylor_series


The variable, a, sets a location. For values of x near a, fewer terms can be used to give a close approximation. However, choosing a = 0 will simplify the sum and give us nice expressions. When a = 0, the Taylor series becomes:


a=0_in_the_Taylor_series


Actually, (x - 0)n is really just xn. Thus, we get the series we call the Maclaurin series.


the_Maclaurin_series


Let's write out a few of the terms in the Maclaurin series:


first_five_terms_of_the_Maclaurin_series


Here are some things to note:

  • f (0) (0) is just the function f(x) with x = 0
  • f (1) (0) is the first derivative of f(x) evaluated at x = 0
  • 0! is ''0 factorial'' and equal to 1
  • 1! is also equal to 1

The first few terms now simplify to:


simplified_first_five_terms


We now need some derivatives. The derivative of cos(x) is -sin(x) and the derivative of sin(x) = cos(x). Also, we'll be evaluating at x = 0. It's good to know cos(0) = 1 and sin(0) = 0. Tabulating this information, we have:

f f(1) f(2) f(3) f(4)
x cos(x) -sin(x) -cos(x) sin(x) cos(x)
x = 0 cos(0) -sin(0) -cos(0) sin(0) cos(0)
simplify 1 0 -1 0 1

Look at those first few terms for the Maclaurin series. The f(x) on the left-hand side is now cos(x). On the right-hand side, substitute from the table:


null


Simplifying:


null


See how the sign alternates between plus and minus? What if we write (-1)n and let n be 0. Then, we have (-1)0 = 1. And what if n = 1? Then, (-1)1 = -1. Now, (-1)n will give us alternating positive and negative signs as n runs from 0, 1, …

And as n runs from 0, 1, … we could consider x2n. This gives us 1, x2, x4, … We can use the same idea for the factorial in the denominator. Putting all of this together, we get a nice compact expression for the Maclaurin series for cos(x):


summation_expression_for_Maclaurin_cosine


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