Find the Exact Value cos(-(3pi)/8)

Problem

cos(−(3*π)/8)

Solution

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

cos(−(3*π)/8)=cos((3*π)/8)

2. Use the half-angle formula for cosine, cos(α/2)=±√(,(1+cos(α))/2) Here, let α/2=(3*π)/8 which means α=(3*π)/4

cos((3*π)/8)=√(,(1+cos((3*π)/4))/2)

3. Determine the sign of the radical. Since (3*π)/8 is in the first quadrant (0<(3*π)/8<π/2, the cosine value must be positive.

cos((3*π)/8)=√(,(1+cos((3*π)/4))/2)

4. Substitute the known value for cos((3*π)/4) which is −√(,2)/2

cos((3*π)/8)=√(,(1−√(,2)/2)/2)

5. Simplify the fraction inside the radical by multiplying the numerator and denominator by 2

cos((3*π)/8)=√(,(2−√(,2))/4)

6. Simplify the square root of the denominator.

cos((3*π)/8)=√(,2−√(,2))/2

Final Answer

cos(−(3*π)/8)=√(,2−√(,2))/2

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