Find the value of cos11π/12?

2 Answers
Jun 12, 2018

-1/4(sqrt2+sqrt6)

Explanation:

"using the "color(blue)"trigonometric identity"

•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)

Jun 12, 2018

- sqrt(2 + sqrt3)/2

Explanation:

cos ((11pi)/12) = cos (-pi/12 + (12pi)/12) = cos (-pi/12 + pi) =
= - cos (-pi/12) = -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 (because cos (pi/12) is positive)
Finally,
cos ((11pi)/12) = - cos (pi/12) = - sqrt(2 + sqrt3)/2
Check by calculator.
cos ((11pi)/12) = cos 165^@ = - 0.966
- sqrt(2 + sqrt3)/2 = - 1.932/2 = - 0.966. Proved