How do you evaluate $arctan (\frac{2}{5})$ $arctan(2/5)$ ?

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How do you evaluate $arctan (\frac{2}{5})$ $arctan(2/5)$ ?

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Answer 1 · 2016-11-27T09:42:29

$arctan (\frac{2}{5}) ≅ \frac{2}{5} - \frac{8}{375} + \frac{32}{15625} ≅ 0.381$ $arctan(2/5) ~=2/5-8/375+32/15625~=0.381$ rounded to the third decimal figure

Explanation:

The mac Laurin series for $arctan (x)$ $arctan(x)$ is

$arctan (x) = x - \frac{x^{3}}{3} + \frac{x^{5}}{5} + \cdot \cdot \cdot + \frac{{(- 1)}^{n}}{2 n + 1} x^{2 n + 1} + o (x^{2 n + 2})$ $arctan(x) = x −x^3/3+x^5/5+ · · · +(−1)^n/(2n+1)x^(2n+1) + o(x^( 2n+2))$

so $arctan (\frac{2}{5}) = \frac{2}{5} - \frac{{(\frac{2}{5})}^{3}}{3} + \frac{{(\frac{2}{5})}^{5}}{5} + 0 ({(\frac{2}{5})}^{6})$ $arctan(2/5)=2/5-(2/5)^3/3+(2/5)^5/5+0((2/5)^6)$

$arctan (\frac{2}{5}) ≅ \frac{2}{5} - \frac{8}{375} + \frac{32}{15625} ≅ 0.381$ $arctan(2/5) ~=2/5-8/375+32/15625~=0.381$ rounded to the third decimal figure

Indeed $tan (0.381) ≅ 0.4 = \frac{2}{5}$ $tan(0.381) ~=0.4=2/5$

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How do you evaluate $arctan (\frac{2}{5})$ $arctan(2/5)$ ?

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