How do you simplify using the half angle formula #cos(-(7pi)/12)#?

1 Answer
Nghi N
Apr 5, 2018

#- sqrt(2 - sqrt3)/2#

Explanation:

#cos ((-7pi)/12) = cos ((-pi)/12 - (6pi)/12) = sin ((-pi)/12) = #
#= - sin ((pi)/12)#
To find #sin ((pi)/12)# use trig half angle identity:
#sin (t/2) = +- sqrt((1 - cos t)/2)#
In this case, #cos t = cos ((2pi)/12) = cos ((pi)/6) = sqrt3/2#
We get, with #(pi/12)# being in Quadrant 1,
#sin ((pi)/12) = sqrt(1 - sqrt3/2)/2 = sqrt(2 - sqrt3)/2#
Finally,
#cos ((-7pi)/12) = - sin (pi/12) = - sqrt(2 - sqrt3)/2#
Check by calculator.
#- sin (pi/12) = - sin 15^@ = - 0.258#
#- sqrt(2 - sqrt3)/2 = - 0.517/2 = - 0.258#. Proved.

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