How do you find the exact functional value tan 7pi/12 using the cosine sum or difference identity?

1 Answer
Sep 26, 2015

Find #tan ((7pi)/12)#

Ans: #(1 + sqrt3)/(1 - sqrt3)#

Explanation:

#sin ((7pi)/12) = sin (pi/3 + pi/4) =#
#= sin (pi/3).cos (pi/4) + sin (pi/4).cos (pi/3) =#
#(sqrt3/2)(sqrt2/2) + (sqrt2/2)(1/2) = ((sqrt2)/4)(1 + sqrt3)#
#cos ((7pi)/12) = cos (pi/3 + pi/4) = #
#= cos (pi/3).cos (pi/4) - sin (pi/3).sin (pi/4) =#
#= (1/2)(sqrt2/2) - (sqrt2/2)(sqrt3/2) = (sqrt2/4)(1 - sqrt3)#

#tan ((7pi)/12) = sin/(cos) = (1 + sqrt3)/(1 - sqrt3)#
Check by calculator:
#tan ((7pi)/12) = tan 105 = -3.73#
#(1 + sqrt3)/(1 - sqr3) = 2.73/(-0.73) = - 3.73#. OK