How do you find the exact value of $cos (arctan (\frac{3}{4}))$ $cos(arctan(3/4))$ ?

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Answer 1 · 2016-12-28T18:15:55

$cos y = \frac{4}{5}$ $cosy=4/5$

Pose:

$y = arctan (\frac{3}{4})$ $y=arctan(3/4)$

this means that:

$tan y = \frac{3}{4}$ $tany=3/4$

Using the trigonometric identity ${sec}^{2} t = 1 + {tan}^{2} t$ $sec^2t=1+tan^2t$ :

$\frac{1}{{cos}^{2} y} = 1 + {tan}^{2} y = 1 + \frac{9}{16} = \frac{25}{16}$ $1/cos^2y = 1+tan^2y = 1+9/16=25/16$

or:

${cos}^{2} y = \frac{16}{25}$ $cos^2y = 16/25$

and:

$cos y = \frac{4}{5}$ $cosy=4/5$

where we take the positive root, since $arctan (\frac{3}{4})$ $arctan(3/4)$ is in the interval $(0, \frac{π}{2})$ $(0,pi/2)$

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