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

1 Answer
Nghi N.
Oct 17, 2015

Find: sec (pi/12)

Explanation:

sec (pi/12) = 1/(cos pi/12). Find cos (pi/12).
Call cos (pi/12) = cos a --> cos 2a = cos ((2pi)/12) = cos (pi/6) = sqrt3/2
Apply the trig identity: cos 2a = 2cos^2 a - 1.
cos 2a = sqrt3/2 = 2cos^2 a - 1
cos^2 a = 1 = sqrt3/2 = (2 + sqrt3)/2
2cos^2 a = (2 + sqrt3)/4
cos a = +- sqrt(2 + sqrt3)/2
Since the arc (pi/12) is in Quadrant I, its cos is positive.
Therefor, cos (pi/12) = cos a = sqrt(2 + sqrt3)/2
sec (pi/12) = 1/(cos a) = 2/sqrt(2 + sqrt3).
Check by calculator:
cos (pi/12) = cos 15 = 0.97
sqrt(2 + sqrt3)/2 = 1.93/2 = 0.97. OK

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