Find the Exact Value cos(pi/8)

Problem

cos(π/8)

Solution

1. Identify the half-angle identity for cosine, which is cos(θ/2)=±√(,(1+cos(θ))/2)

2. Substitute θ=π/4 into the identity, since π/8=π/4/2

3. Determine the sign of the result. Since π/8 is in the first quadrant, the cosine value must be positive.

4. Evaluate cos(π/4) which is √(,2)/2

5. Substitute the value into the formula:

cos(π/8)=√(,(1+√(,2)/2)/2)

6. Simplify the expression inside the radical by finding a common denominator in the numerator:

cos(π/8)=√(,(2+√(,2))/2/2)

7. Divide by 2 to reach the final simplified form:

cos(π/8)=√(,(2+√(,2))/4)

8. Extract the square root of the denominator:

cos(π/8)=√(,2+√(,2))/2

Final Answer

cos(π/8)=√(,2+√(,2))/2

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