How do you evaluate Sin ((13pi)/12)?

1 Answer
Nghi N.
Mar 24, 2016

- sqrt(2 - sqrt3)/2

Explanation:

sin (13pi)/12 = sin (pi/12 + pi) = - sin (pi/12).
Evaluate sin (pi/12) by the trig identity:
cos 2a = 1 - 2sin^2 a
Call pi/12) = t, we get cos (2t) = cos (pi/6) = sqrt3/2
The equation becomes:
sqrt3/2 = 1 - 2 sin^2 t
2sin^2 t = 1 - sqrt3/2 = (2 - sqrt3)/2
sin^2 t = (2 - sqrt3)/4
sin t = sin (pi/12) = +- sqrt(2 - sqrt3)/2
Because sin (pi/12) is positive, therefor,
sin ((13pi)/12) = - sin (pi/12) is negative.
Answer: sin ((13pi)/12) = - (sqrt(2 - sqrt3))/2

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