# 2arctan(1/2)+arctan(1/7)=?

Jun 24, 2018

$2 a r c \tan \left(\frac{1}{2}\right) + a r c \tan \left(\frac{1}{7}\right) = a r c \tan \left(\frac{31}{17}\right)$

#### Explanation:

We know that ,

color(red)((1)arc tanx+arc tany=arc tan((x+y)/(1-x*y)) , color(red)(x*y < 1

Let , $A = 2 a r c \tan \left(\frac{1}{2}\right) + a r c \tan \left(\frac{1}{7}\right)$

Using $\left(1\right)$ we get ,

$\textcolor{b l u e}{2 a r c \tan \left(\frac{1}{2}\right)} = a r c \tan \left(\frac{1}{2}\right) + a r c \tan \left(\frac{1}{2}\right)$

$\textcolor{w h i t e}{2 a r c \tan \left(\frac{1}{2}\right)} = a r c \tan \left(\frac{\frac{1}{2} + \frac{1}{2}}{1 - \frac{1}{2} \cdot \frac{1}{2}}\right) ,$$x \cdot y = \frac{1}{2} \cdot \frac{1}{2}$=$\frac{1}{4} < 1$

$\textcolor{w h i t e}{2 a r c \tan \left(\frac{1}{2}\right)} = a r c \tan \left(\frac{1}{1 - \frac{1}{4}}\right)$

$\textcolor{w h i t e}{2 a r c \tan \left(\frac{1}{2}\right)} = a r c \tan \left(\frac{1}{\frac{3}{4}}\right)$

color(blue)(2arc tan(1/2)=arc tan(4/3)

So,

A=color(blue)(arc tan (4/3))+arc tan(1/7)tocolor(red)( Apply(1)

$A = a r c \tan \left(\frac{\frac{4}{3} + \frac{1}{7}}{1 - \frac{4}{3} \cdot \frac{1}{7}}\right) , x \cdot y = \frac{4}{3} \cdot \frac{1}{7} = \frac{4}{21} < 1$

$A = a r c \tan \left(\frac{28 + 3}{21 - 4}\right)$

$A = a r c \tan \left(\frac{31}{17}\right)$

Jun 24, 2018

${61}^{\circ} 26$

#### Explanation:

Use calculator:
$\tan x = \frac{1}{2}$ --> arc $x = {26}^{\circ} 57$
$2 a r c x = 2 \left(26.57\right) = {53}^{\circ} 13$
$\tan y = \frac{1}{7}$ --> arc $y = {8}^{\circ} 13$
$2 \arctan \left(\frac{1}{2}\right) + \arctan \left(\frac{1}{7}\right) = {53}^{\circ} 13 + {8}^{\circ} 13 = {61}^{\circ} 26$