find value of tan46^0 using calculus


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Question Number 118004 by TANMAY PANACEA last updated on 14/Oct/20

$f i n d v a l u e o f t a n 46^{0} u s i n g c a l c u l u s$
	Answered by mr W last updated on 14/Oct/20

	$46 ° = 45 ° + 1 ° = \frac{π}{4} + \frac{π}{180}$ $\tan 46 ° = \frac{1 + \tan \frac{π}{180}}{1 - \tan \frac{π}{180}}$ $\approx \frac{1 + \frac{π}{180}}{1 - \frac{π}{180}} = \frac{180 + π}{180 - π} = 1.0355$ $o r$ $y = \tan x$ $\frac{Δ y}{Δ x} \approx \frac{d y}{d x} = \frac{1}{\cos^{2} x}$ $Δ y = \tan 46 ° - \tan 45 ° \approx \frac{Δ x}{\cos^{2} 45 °}$ $\tan 46 ° \approx \tan 45 ° + \frac{1}{\cos^{2} 45 °} \times \frac{π}{180}$ $\approx 1 + 2 \times \frac{π}{180} = 1 + \frac{π}{90} = 1.0349$
	Commented by TANMAY PANACEA last updated on 14/Oct/20

	$t h a n k y o u s i r$
	Commented by TANMAY PANACEA last updated on 14/Oct/20

	$s i r e n c o u r a g e p e o p l e t o p o s t l o g i c b a s e d$ $a n d i n t e r e s t i n g q u e s t i o n .$ $p e o p l e a r e p o s t i n g M . S c l e v e l t e x t b o o k q u e s t i o n$
	Answered by TANMAY PANACEA last updated on 14/Oct/20

	$y = t a n x \to \frac{d y}{d x} = {s e c}^{2} x$ $\frac{d y}{d x} \approx \frac{△ y}{△ x}$ $h e r e x = 45^{o} = \frac{π}{4}, x + △ x = 46^{o} = \frac{π}{4} + \frac{π}{180} = \frac{46 π}{180}$ $n o w w h e n x = \frac{π}{4} y = t a n x = 1$ $w h e n x + △ x = \frac{46 π}{180} y + △ y = ?$ $h e r e y = t a n x = t a n 45^{o} = 1$ $△ y = \frac{d y}{d x} \times △ x = {s e c}^{2} x \times △ x$ $△ y = {s e c}^{2} (\frac{π}{4}) \times \frac{π}{180} = \frac{π}{90}$ $y + △ y = 1 + \frac{π}{90}$
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