Question #f29c8

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Question #f29c8

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Answer 1 · 2017-01-08T18:10:44

$y = π + 4 k π, k = 0, 1, 2, \dots$ $y = pi+4kpi, k = 0,1,2,cdots$

Explanation:

$tan (a + b) = \frac{tan a + tan b}{1 - tan a tan b}$ $tan(a+b)=(tan a+tanb)/(1-tan a tanb)$

so calling

$\frac{y}{4} = 2 arctan (\frac{1}{3}) + arctan (\frac{1}{7})$ $y/4 = 2arctan(1/3)+arctan(1/7)$ then

$tan (\frac{y}{4}) = tan (2 arctan (\frac{1}{3}) + arctan (\frac{1}{7})) = \frac{\frac{3}{4} + \frac{1}{7}}{1 - \frac{3}{4} \frac{1}{7}} = 1$ $tan(y/4)=tan(2arctan(1/3)+arctan(1/7))=(3/4+1/7)/(1-3/4 1/7)=1$

so

$tan (\frac{y}{4}) = tan (\frac{π}{4} + k π)$ $tan(y/4)=tan(pi/4+kpi)$ then

$y = π + 4 k π, k = 0, 1, 2, \dots$ $y = pi+4kpi, k = 0,1,2,cdots$

Answer 2 · 2017-01-08T19:03:13

$180^{\circ}$ $180^@$

Explanation:

Use calculator.
$arctan (\frac{1}{3}) = 18^{\circ} 43$ $arctan (1/3) = 18^@43$
$2 arctan (\frac{1}{3}) = 36^{\circ} 87$ $2arctan (1/3) = 36^@87$
$arctan (\frac{1}{7}) = 8^{\circ} 13$ $arctan (1/7) = 8^@13$
$2 arctan (\frac{1}{3}) + arctan (\frac{1}{7}) = 36.87 + 8.13 = 45^{\circ}$ $2arctan (1/3) + arctan (1/7) = 36.87 + 8.13 = 45^@$
Finally,
$4 (2 arctan (\frac{1}{3}) + arctan (\frac{1}{7})) = 4 (45^{\circ}) = 180^{\circ}$ $4(2arctan (1/3) + arctan (1/7)) = 4 (45^@) = 180^@$

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