How do you find cos 11pi/12?

2 Answers
Jim G.
Jun 21, 2018

#-1/4(sqrt2+sqrt6)#

Explanation:

#"using the "color(blue)"addition formula for cos"#

#•color(white)(x)cos(x+y)=cosxcosy-sinxsiny#

#"note that "(11pi)/12=(2pi)/3+pi/4#

#cos((11pi)/12)=cos((2pi)/3+pi/4)#

#=cos((2pi)/3)cos(pi/4)-sin((2pi)/3)sin(pi/4)#

#=-cos(pi/3)cos(pi/4)-sin(pi/3)sin(pi/4)#

#=(-1/2xxsqrt2/2)-(sqrt3/2xxsqrt2/2)#

#=-sqrt2/4-sqrt6/4#

#=-1/4(sqrt2+sqrt6)#

Nghi N.
Jun 22, 2018

#cos ((11pi)/12) = - sqrt(2 + sqrt3)/2#

Explanation:

#cos ((11pi)/12) = cos (-pi/12 + pi) = cos (pi/12 + pi) = - cos (pi/12)#
Find #cos (pi/12)# by using trig identity:
#2cos^2 a = 1 + cos 2a#
In this case,
#2cos^2 (pi/12) = 1 + cos (pi/6) = 1 + sqrt3/2 = (2 + sqrt3)/2#
#cos^2 (pi/12) = (2 + sqrt3)/4#
#cos (pi/12) = sqrt(2 + sqrt3)/2# (#cos (pi/12)# is positive)
Finally,
#cos ((11pi)/12) = - cos (pi/12) = - sqrt(2 + sqrt3)/2#

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