How do you use the sum and difference formula to simplify #sec((17pi)/12)#?

1 Answer
Nghi N.
Feb 6, 2017

#- csc (pi/12)#

Explanation:

First, simplify cos ((17pi)/12)
Use unit circle, property of complements and supplement arcs:
#cos ((17pi)/12) = cos ((5pi)/12 + pi) = - cos ((5pi)/12)#
#cos ((5pi)/12) = cos (-pi/12 + (6pi)/12) = cos (-pi/12 + pi/2) = sin (pi/2)#.
Finally,
#cos ((17pi)/12) = - cos ((5pi)/12) = - sin (pi/12)#
#sec ((17pi)/12) = - 1/sin (pi/12) = - csc (pi/12)#
Check by calculator:
#cos ((17pi)/12) = cos 255^@ = -0.259 --> sec = - 1/0.259 = -3.86 #
# - sin (pi/12) = - sin 15^@ = - 0.259 -> - csc (pi/12) = - 3.86 #. OK

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