Question

How do you evaluate $ \arctan \left( {\dfrac{2}{5}} \right) $

Answer 1

Hint: Here this question is related to the inverse trigonometry concept. The inverse is also called an arc. Here arctan means inverse of tangent. We solve this question by using the maclaurin series expansion for $ \arctan \left( x \right) $ , here the value of x is $ \left( {\dfrac{2}{5}} \right) $ and we determine the value for the $ \arctan \left( {\dfrac{2}{5}} \right) $

Complete step-by-step answer:
The question is about the inverse trigonometry. In this question we have to find the inverse value of tangent trigonometry ratio. The word arc means inverse, the inverse is also called as arc. To solve $ \arctan \left( {\dfrac{2}{5}} \right) $ we use the maclaurin series expansion. The maclaurin series expansion for $ \arctan \left( x \right) $ is given by $ \arctan \left( x \right) = x - \dfrac{{{x^3}}}{3} + \dfrac{{{x^5}}}{5} + ... $ the value of x is $ \left( {\dfrac{2}{5}} \right) $
So now we will substitute the value of x and we get
 $ \Rightarrow \arctan \left( {\dfrac{2}{5}} \right) = \dfrac{2}{5} - \dfrac{{{{\left( {\dfrac{2}{5}} \right)}^3}}}{3} + \dfrac{{{{\left( {\dfrac{2}{5}} \right)}^5}}}{5} $
We will consider only three terms and other terms will be smaller and smaller so we are neglecting it. We can neglect the other terms; it does not make any change in obtaining the solution for the question. We simplify the numerator terms
 $ \Rightarrow \arctan \left( {\dfrac{2}{5}} \right) \cong \dfrac{2}{5} - \dfrac{8}{{375}} + \dfrac{{32}}{{15625}} $
Applying the division to each term we get
 $ \Rightarrow \arctan \left( {\dfrac{2}{5}} \right) \cong 0.4 - 0.02133 + 0.00205 $
By applying the addition and subtraction operations we get
 $ \Rightarrow \arctan \left( {\dfrac{2}{5}} \right) \cong 0.381 $
The value of $ \arctan \left( {\dfrac{2}{5}} \right) $ is approximately equal to 0.381.
Hence, we got the value of $ \arctan \left( {\dfrac{2}{5}} \right) $ by using the maclaurin series expansion for $ \arctan \left( x \right) $
We can solve the value of $ \arctan \left( {\dfrac{2}{5}} \right) $ by the direct method or by using the calculator and hence we get the same value.
So, the correct answer is “ 0.381 ”.

Note: The trigonometry and inverse trigonometry are inverse for each other. The inverse of a function is represented as the arc of the function or the function is raised by the power -1. For the trigonometry and the inverse trigonometry we need to know about the table of trigonometry ratios for the standard angles. for some angels we need to know the maclurin’s series expansion