Find the Exact Value cos((5pi)/8)

Problem

cos((5*π)/8)

Solution

1. Identify the angle as half of a known angle on the unit circle. Since (5*π)/8=1/2⋅(5*π)/4 we can use the half-angle formula for cosine.

2. State the half-angle formula for cosine.

cos(θ/2)=±√(,(1+cos(θ))/2)

3. Determine the sign based on the quadrant. The angle (5*π)/8 is in the second quadrant (90<112.5<180, where cosine is negative.

cos((5*π)/8)=−√(,(1+cos((5*π)/4))/2)

4. Substitute the known value cos((5*π)/4)=−√(,2)/2 into the formula.

cos((5*π)/8)=−√(,(1−√(,2)/2)/2)

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

cos((5*π)/8)=−√(,(2−√(,2))/2/2)

6. Divide by 2 to simplify the fraction further.

cos((5*π)/8)=−√(,(2−√(,2))/4)

7. Extract the square root of the denominator.

cos((5*π)/8)=−√(,2−√(,2))/2

Final Answer

cos((5*π)/8)=−√(,2−√(,2))/2

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