Find the Exact Value cos((7pi)/8)

Problem

cos((7*π)/8)

Solution

1. Identify the angle as a half-angle. Since (7*π)/8=1/2⋅(7*π)/4 we can use the half-angle identity for cosine.

2. Recall the half-angle formula for cosine.

cos(θ/2)=±√(,(1+cos(θ))/2)

3. Determine the sign of the result. The angle (7*π)/8 is in the second quadrant (90<157.5<180, where cosine is negative.

cos((7*π)/8)=−√(,(1+cos((7*π)/4))/2)

4. Evaluate the cosine of the reference angle. The angle (7*π)/4 is in the fourth quadrant with a reference angle of π/4

cos((7*π)/4)=√(,2)/2

5. Substitute the value into the formula.

cos((7*π)/8)=−√(,(1+√(,2)/2)/2)

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

cos((7*π)/8)=−√(,(2+√(,2))/4)

7. Apply the square root to the denominator.

cos((7*π)/8)=−√(,2+√(,2))/2

Final Answer

cos((7*π)/8)=−√(,2+√(,2))/2

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