Evaluate cos((3pi)/8)

Problem

cos((3*π)/8)

Solution

1. Identify the angle as a half-angle of a known value. Since (3*π)/8=1/2⋅(3*π)/4 we can use the half-angle identity for cosine.

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

3. Determine the sign based on the quadrant. The angle (3*π)/8 is in the first quadrant (0<(3*π)/8<π/2, so the cosine value is positive.

4. Substitute θ=(3*π)/4 into the formula.

cos((3*π)/8)=√(,(1+cos((3*π)/4))/2)

5. Evaluate the cosine of the reference angle. We know cos((3*π)/4)=−√(,2)/2

cos((3*π)/8)=√(,(1−√(,2)/2)/2)

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

cos((3*π)/8)=√(,(2−√(,2))/2/2)

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

cos((3*π)/8)=√(,(2−√(,2))/4)

cos((3*π)/8)=√(,2−√(,2))/2

Final Answer

cos((3*π)/8)=√(,2−√(,2))/2

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