Find the Value Using the Unit Circle cos((5pi)/8)

Problem

cos((5*π)/8)

Solution

1. Identify the angle (5*π)/8 and recognize that it is not a standard angle on the unit circle, but it is half of the standard angle (5*π)/4

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

3. Substitute θ=(5*π)/4 into the formula. Since (5*π)/8 is in the second quadrant (between π/2 and π, the cosine value must be negative.

cos((5*π)/8)=−√(,(1+cos((5*π)/4))/2)

4. Evaluate the cosine of the standard angle (5*π)/4 which is −√(,2)/2

cos((5*π)/8)=−√(,(1−√(,2)/2)/2)

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

cos((5*π)/8)=−√(,(2−√(,2))/2/2)

6. Divide by 2 to reach the final radical form.

cos((5*π)/8)=−√(,(2−√(,2))/4)

7. Simplify the radical by taking 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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