How do I find the value of csc pi/12?

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How do I find the value of csc pi/12?

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Answer 1 · 2015-08-12T16:58:26

Find $csc (pi/12)$

Ans: $2/sqrt(2 - sqrt3)$

Explanation:

$csc (pi/12) = 1/sin (pi/12)$ . Find $sin (pi/12)$
$call sin (pi/12) = sin t$
$cos 2t = cos ((2pi)/12) = cos ((pi)/6) = sqrt3/2.$
Use trig identity: $cos 2t = sqrt3/2 = 1 - 2sin^2 t$
$2sin^2 t = 1 - sqrt3/2$ --> $sin^2 t = (2 - sqrt3)/4$
$sin (pi/12) = sin t = +- sqrt(2 - sqrt3)/2$
Since sin $(pi/12)$ is in Quadrant I, therefor, only the positive answer is accepted.

$csc (pi/12) = 2/(sqrt(2 - sqrt3)$

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How do I find the value of csc pi/12?

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