How to Find Series Expansion of the Inverse Trigonometric Functions: A Novel Method
Trigonometric functions are one of the most important mathematical functions which appear almost in all branches of physics and mathematics. The inverses of trigonometric functions are often required in various kinds of problems. In this article I will discuss a novel method of finding the series expansion of the inverse trigonometric functions.
The Problem with Taylor Expansion
The mostly used method of series expansion is to obtain the Taylor expansion of the given function near a suitable value of the argument. But, in case of inverse trigonometric functions it is quite difficult because of the fact that higher order derivatives of the inverse trigonometric functions become extremely complicated involving too many terms and large indices.
The method described here requires very basic knowledge of determining integration, derivatives and binomial expansion. For the case of ArcSin(x), ArcCos(x) and ArcTan(x) the steps are as follows.
- Find the derivative of the inverse trigonometric function.
- Expand the expression of derivative using binomial expansion.
- Perform integration over the expansion term by term.
- Choose proper integration constant to get the final result.
For the cases of the rest three functions we will simply replace the argument x by its reciprocal 1/x to obtain the series expansions. The detailed steps are illustrated below by considering each of them one by one.
Note: Here the inverse functions are described by using the prefix “Arc”.
1. Expansion of ArcTan(x)
Let us first consider the inverse of Tan(x) which has the simplest form of series expansion. Our first step is to find the derivative of ArcTan(x), which is 1/(1+x^2). Since integration is the inverse operation of derivative, ArcTan(x) will be same as the integral of 1/(1+x^2). We will perform the integral, after doing the binomial expansion, term by term. This will give the expansion with an additive integration constant term which will be zero in this case, since ArcTan(x) is zero at x=0. While doing the binomial expansion, it is assumed that |x|<1. All the steps are shown below.
2. Expansion of ArcSin(x)
The derivative of ArcSin(x) is 1/Sqrt(1-x^2). We will first expand it binomially and then integrate it term by term. The integration constant, in this case too, becomes zero. The details are shown below.
3. Expansion of ArcCos(x)
It can be done in two ways. First by following the same procedure remembering that derivative of ArcCos(x) is (-1)/Sqrt(1-x^2). In this case the integration constant becomes Pi/2 since ArcCos(x)=Pi/2 at x=0.
The alternative way is to simply use the identity that ArcCos(x) = Pi/2 – ArcSin(x), which will lead to the same result.
4. Expansion of ArcSec(x), ArcCsc(x) and ArcCot(x)
Since Sec(x), Csc(x) and Cot(x) are reciprocal of Cos(x), Sin(x) and Tan(x) respectively, the expansion of them are easily obtained by replacing x by 1/x.
Thus we see that the series expansion of the inverse trigonometric functions can be obtained easily using mere integration without doing the cumbersome Taylor expansion.
Note: If you are unable to view the images clearly, please zoom in the page.