Find the Exact Value cos(pi/4-pi/6)

Problem

cos(π/4−π/6)

Solution

1. Identify the appropriate trigonometric identity. We use the cosine difference formula:

cos(α−β)=cos(α)*cos(β)+sin(α)*sin(β)

2. Substitute the values α=π/4 and β=π/6 into the formula:

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

3. Evaluate the trigonometric functions for the known special angles:

cos(π/4)=√(,2)/2

cos(π/6)=√(,3)/2

sin(π/4)=√(,2)/2

sin(π/6)=1/2

4. Multiply the terms:

cos(π/4−π/6)=(√(,2)/2)*(√(,3)/2)+(√(,2)/2)*(1/2)

cos(π/4−π/6)=√(,6)/4+√(,2)/4

5. Combine the fractions over a common denominator:

cos(π/4−π/6)=(√(,6)+√(,2))/4

Final Answer

cos(π/4−π/6)=(√(,6)+√(,2))/4

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