How do you find the exact value of $sec (arctan (\frac{1}{2}))$ $Sec(Arctan (1/2))$ ?

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How do you find the exact value of $sec (arctan (\frac{1}{2}))$ $Sec(Arctan (1/2))$ ?

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Answer 1 · 2016-04-27T04:26:56

$\frac{2}{\sqrt{5}}$ $2/sqrt 5$

Explanation:

Let a = arc tan (1/2). Then $tan a = \frac{1}{2} . S o, sec a = \frac{2}{\sqrt{5}}$ $tan a = 1/2. So, sec a = 2/sqrt 5$ , using $sec a = \pm \sqrt{1 + {tan}^{2} a} = \pm \sqrt{1 + \frac{1}{4}} = \pm \frac{2}{\sqrt{5}}$ $sec a = +-sqrt ( 1 + tan^2a)=+-sqrt ( 1+ 1/4) = +-2/sqrt 5$ . Here, tan a is positive. So, sec a is positive and $= \frac{2}{\sqrt{5}}$ $= 2/sqrt 5$

Now, $sec (a r c tan (\frac{1}{2})) = sec (a r c sec (\frac{2}{\sqrt{5}})) = \frac{2}{\sqrt{5}}$ $sec(arc tan (1/2))=sec(arc sec(2/sqrt 5))=2/sqrt5$

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How do you find the exact value of $sec (arctan (\frac{1}{2}))$ $Sec(Arctan (1/2))$ ?

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