How do you use the sum and difference identities to find the exact values of the sine, cosine and tangent of the angle #(7pi)/12#?

1 Answer
Jul 17, 2015

Find 3 trig functions of #((7pi)/12)#

Explanation:

Call #sin ((7pi)/12) = sin t# -> # cos 2t = cos ((14pi)/12)#
On the trig unit circle.
#cos ((14pi)/12) = cos (pi/6 + pi) = - cos (pi/6) = -(sqrt3)/2#
Apply the trig identity: #cos 2t = 1 - 2sin^2 t#

#- (sqrt3)/2 = 1 - 2sin^2 t#
#sin^2 t = (2 + sqrt3)/4#
#sin t = sin ((7pi)/12) = +- sqrt(2 + sqrt3)/2#

Apply trig identity: #cos 2t = 2cos^2 t - 1#
#-(sqrt3)/2 = 2cos^2 t - 1#
# 2cos^2 t = 1 - (sqrt3)/2 = (2 - sqrt3)/2#
#cos^2 t = (2 - sqrt3)/4#
#cos t = cos ((7pi)/12) = +- sqrt(2 - sqrt3)/2#

#tan ((7pi)/12) = sin t/cos t = +- (sqrt2 +sqrt3)/sqrt(2 - sqrt3)#