First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, π/12 can be split into π/3−π/4.
cos(π/3−π/4)
Use the difference formula for cosine to simplify the expression. The formula states that cos(A−B)=cos(A)cos(B)+sin(A)sin(B)
cos(π/3)⋅cos(π/4)+sin(π/3)⋅sin(π/4)
The exact value of cos(π/3) is 12, so:
(12)⋅cos(π/4)+sin(π/3)⋅sin(π/4)
The exact value of cos(π/4) is √22.
(12)⋅(√22)+sin(π/3)⋅sin(π/4)
The exact value of sin(π/3) is √32.
(12)⋅(√22)+(√32)⋅sin(π/4)
The exact value of sin(π/4) is √22.
(12)⋅(√22)+(√32)⋅(√22)
Simplify each term:
√24+√64
Combine the numerators over the common denominator.
(√2+√6) / 4