Find the Exact Value cos((23pi)/12)

Problem

cos((23*π)/12)

Solution

1. Identify the angle as being in the fourth quadrant. Since (23*π)/12 is close to 2*π (which is (24*π)/12, we can rewrite the angle using a reference angle or as a sum/difference of known angles.

2. Rewrite the expression using the difference of two common angles:

(23*π)/12=2*π−π/12

3. Apply the periodicity property cos(2*π−θ)=cos(θ)

cos((23*π)/12)=cos(π/12)

4. Decompose the angle π/12 into the difference of two standard angles:

π/12=π/3−π/4

5. Apply the cosine difference identity cos(A−B)=cos(A)*cos(B)+sin(A)*sin(B)

cos(π/3−π/4)=cos(π/3)*cos(π/4)+sin(π/3)*sin(π/4)

6. Substitute the exact values for the trigonometric functions:

cos(π/3)=1/2

cos(π/4)=√(,2)/2

sin(π/3)=√(,3)/2

sin(π/4)=√(,2)/2

7. Simplify the resulting expression:

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

(√(,2)+√(,6))/4

Final Answer

cos((23*π)/12)=(√(,6)+√(,2))/4

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