How do you find the exact functional value tan 165 using the cosine sum or difference identity?

2 Answers
Sep 21, 2015

#sqrt(2-sqrt3)/-sqrt(2+sqrt3)#

Explanation:

#sin(165)=sin(180-15)=sin(15)=sqrt((1-cos2(15))/2)#=#sqrt((1-cos30)/2)=sqrt((1-sqrt3/2)/2)=sqrt(2-sqrt3)/2#
#cos(165)=cos(180-15)=-cos(15)=-sqrt((1+cos2(15))/2#=#-sqrt((1+cos30)/2)=-sqrt((1+sqrt3/2)/2)=-sqrt(2+sqrt3)/2#
#tan(165)=sin(165)/cos(165)=sqrt(2-sqrt3)/2/-sqrt(2+sqrt3)/2=sqrt(2-sqrt3)/-sqrt(2+sqrt3)#

Sep 21, 2015

Find tan (165)

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

Explanation:

Apply the trig identity: #tan (a + b) = (tan a + tan b)/(1 - tan a.tan b)#
tan (165) = tan (45 + 120).
Trig table gives --> tan 45 = 1; #tan 120 = -sqrt3#

#tan (165) = (1 - sqrt3)/(1 + sqrt3)#

Check by calculator: tan (165) = - 0.27
#(1 - sqrt3)/(1 + sqrt3) = - 0.73/2.73 = - 0.27#. OK