Question #c5acc

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Question #c5acc

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Answer 1 · 2017-05-26T04:08:42

$\frac{\sqrt{2 - \sqrt{3}}}{2}$ $sqrt(2 - sqrt3)/2$

Explanation:

$cos (\frac{3 π}{4} - \frac{π}{3}) = cos (\frac{9 π - 4 π}{12}) = cos (\frac{5 π}{12})$ $cos ((3pi)/4 - pi/3) = cos ((9pi - 4pi)/12) = cos ((5pi)/12)$
Find $cos (\frac{5 π}{12})$ $cos ((5pi)/12)$ by using trig identity:
$2 {cos}^{2} a = 1 + cos 2 a$ $2cos^2 a = 1 + cos 2a$
In this case, trig table gives:
$cos 2 a = cos (\frac{10 π}{12}) = cos (5 \frac{π}{6}) = - \frac{\sqrt{3}}{2}$ $cos 2a = cos ((10pi)/12) = cos (5pi/6) = - sqrt3/2$
Call $cos (\frac{5 π}{12}) = cos t$ $cos ((5pi)/12) = cos t$ , we get:
$2 {cos}^{2} t = 1 - \frac{\sqrt{3}}{2} = \frac{2 - \sqrt{3}}{2}$ $2cos^2 t = 1 - sqrt3/2 = (2 - sqrt3)/2$
${cos}^{2} t = \frac{2 - \sqrt{3}}{4}$ $cos^2 t = (2 - sqrt3)/4$
$cos t = \pm \frac{\sqrt{2 - \sqrt{3}}}{2}$ $cos t = +- sqrt(2 - sqrt3)/2$
Since $\frac{5 π}{12}$ $(5pi)/12$ is in Quadrant 1, its cos is positive -->
$cos (\frac{5 π}{12}) = cos t = \frac{\sqrt{2 - \sqrt{3}}}{2}$ $cos ((5pi)/12) = cos t = sqrt(2 - sqrt3)/2$

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Question #c5acc

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