Expand Using Sum/Difference Formulas cos(pi/12)

Problem

cos(π/12)

Solution

1. Identify two common angles from the unit circle whose difference is π/12

π/12=(4*π)/12−(3*π)/12

π/12=π/3−π/4

2. Apply the formula for the cosine of a difference, which is 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)

3. Substitute the exact values for the trigonometric functions of the special angles.

cos(π/3)=1/2

cos(π/4)=√(,2)/2

sin(π/3)=√(,3)/2

sin(π/4)=√(,2)/2

4. Multiply the terms together.

(1/2)*(√(,2)/2)=√(,2)/4

(√(,3)/2)*(√(,2)/2)=√(,6)/4

5. Simplify by adding the fractions.

√(,2)/4+√(,6)/4=(√(,2)+√(,6))/4

Final Answer

cos(π/12)=(√(,2)+√(,6))/4

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