$2 arctan (\frac{1}{2}) + arctan (\frac{1}{7}) =$ $2arctan(1/2)+arctan(1/7)=$ ?

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$2 arctan (\frac{1}{2}) + arctan (\frac{1}{7}) =$ $2arctan(1/2)+arctan(1/7)=$ ?

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Answer 1 · 2018-06-24T16:32:40

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

Explanation:

We know that ,

$(1) a r c tan x + a r c tan y = a r c tan (\frac{x + y}{1 - x \cdot y})$ $color(red)((1)arc tanx+arc tany=arc tan((x+y)/(1-x*y))$ , $x \cdot y < 1$ $color(red)(x*y < 1$

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

Using $(1)$ $(1)$ we get ,

$2 a r c tan (\frac{1}{2}) = a r c tan (\frac{1}{2}) + a r c tan (\frac{1}{2})$ $color(blue)(2arc tan(1/2))=arc tan(1/2)+arc tan(1/2)$

$2 a r c tan (\frac{1}{2}) = a r c tan (\frac{\frac{1}{2} + \frac{1}{2}}{1 - \frac{1}{2} \cdot \frac{1}{2}}),$ $color(white)(2arc tan(1/2))=arc tan((1/2+1/2)/(1-1/2*1/2)),$ $x \cdot y = \frac{1}{2} \cdot \frac{1}{2}$ $x*y=1/2*1/2$ = $\frac{1}{4} < 1$ $1/4<1$

$2 a r c tan (\frac{1}{2}) = a r c tan (\frac{1}{1 - \frac{1}{4}})$ $color(white)(2arc tan(1/2))=arc tan(1/(1-1/4))$

$2 a r c tan (\frac{1}{2}) = a r c tan (\frac{1}{\frac{3}{4}})$ $color(white)(2arc tan(1/2))=arc tan(1/(3/4))$

$2 a r c tan (\frac{1}{2}) = a r c tan (\frac{4}{3})$ $color(blue)(2arc tan(1/2)=arc tan(4/3)$

So,

$A = a r c tan (\frac{4}{3}) + a r c tan (\frac{1}{7}) \to A p p l y (1)$ $A=color(blue)(arc tan (4/3))+arc tan(1/7)tocolor(red)( Apply(1)$

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

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

$A = a r c tan (\frac{31}{17})$ $A=arc tan(31/17)$

Answer 2 · 2018-06-24T18:52:08

$61^{\circ} 26$ $61^@26$

Explanation:

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

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