Chapter 8.6, Problem 31E

To determine

To find: The value of $\arctan 0.2$ correct to five decimal places.

Answer to Problem 31E

Therequired value of arctan0.2 is $0.19739$ .

Explanation of Solution

Calculation:

Consider the example 7 shows that,

  $\begin{array}{c}\arctan x={\tan }^{-1}x\\ =x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}\\ ={\displaystyle \sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n{x}^{2n+1}}{2n+1}}\end{array}$

Use alternate series estimation with $x=0.2$ then,

  $\begin{array}{l}{b}_1=0.2\\ b_2=2.667\times {10}^{-3}\\ b_3=6.4\times {10}^{-5}\\ b_4=1.829\times {10}^{-6}\cdot {10}^{-7}\end{array}$

The above error is so small it won’t affect the approximation of the fifth decimal digit.

Consider the alternate series estimation theorem states that $\left|s-s_n\right|\le b_{n+1}$ . Then with the $n+1=4$ , the value of $s_3$ is,

  $\begin{array}{c}{s}_3=0.2-\frac{{0.2}^3}{3}+\frac{{0.2}^5}{5}\\ =0.19739\end{array}$