Question #e4b32

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Question #e4b32

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Answer 1 · 2017-03-06T01:33:55

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

Explanation:

Use unit circle, trig table and property of complement arcs.
$sin (\frac{5 π}{12}) = sin (\frac{π}{12} + \frac{π}{2}) = cos (\frac{π}{12})$ $sin ((5pi)/12) = sin (pi/12 + pi/2) = cos (pi/12)$ .
Evaluate $cos (\frac{π}{12})$ $cos (pi/12)$ by using trig identity:
$2 {cos}^{2} a = 1 + cos 2 a$ $2cos^2 a = 1 + cos 2a$
In this case,
$2 {cos}^{2} (\frac{π}{12}) = 1 + cos (\frac{π}{6}) = 1 + \frac{\sqrt{3}}{2} = \frac{2 + \sqrt{3}}{2}$ $2cos^2 (pi/12) = 1 + cos (pi/6) = 1 + sqrt3/2 = (2 + sqrt3)/2$
${cos}^{2} (\frac{π}{12}) = \frac{2 + \sqrt{3}}{4}$ $cos^2 (pi/12) = (2 + sqrt3)/4$
$cos (\frac{π}{12}) = \pm \frac{\sqrt{2 + \sqrt{3}}}{2}$ $cos (pi/12) = +- sqrt(2 + sqrt3)/2$ .
Because $sin (\frac{π}{12})$ $sin (pi/12)$ is positive, take the positive value.
Finally,
$sin (\frac{5 π}{12}) = cos (\frac{π}{12}) = \frac{\sqrt{2 + \sqrt{3}}}{2}$ $sin ((5pi)/12) = cos (pi/12) = sqrt(2 + sqrt3)/2$

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