Find the Exact Value cos((-pi)/12)

Problem

cos(−π/12)

Solution

1. Apply the even-odd identity for the cosine function, which states that cos(−θ)=cos(θ)

cos(−π/12)=cos(π/12)

2. Rewrite the angle as a difference of two common angles from the unit circle, such as π/4 and π/6

π/12=(3*π)/12−(2*π)/12=π/4−π/6

3. Apply the cosine difference formula, cos(A−B)=cos(A)*cos(B)+sin(A)*sin(B) where A=π/4 and B=π/6

cos(π/4−π/6)=cos(π/4)*cos(π/6)+sin(π/4)*sin(π/6)

4. Substitute the exact values for the sine and cosine of the common angles.

cos(π/4)=√(,2)/2

cos(π/6)=√(,3)/2

sin(π/4)=√(,2)/2

sin(π/6)=1/2

5. Multiply and simplify the resulting expression to find the final value.

√(,2)/2⋅√(,3)/2+√(,2)/2⋅1/2=√(,6)/4+√(,2)/4

Final Answer

cos(−π/12)=(√(,6)+√(,2))/4

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