Question

Find the value of cos11π/12?

Answer 1

`-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)`

Answer 2

`- 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