Simplify cos((9pi)/8)

Problem

cos((9*π)/8)

Solution

1. Identify the angle in terms of a reference angle. Since (9*π)/8=π+π/8 the angle is in the third quadrant.

2. Apply the quadrant rule for cosine. In the third quadrant, cosine is negative, so cos(π+θ)=−cos(θ)

3. Substitute the reference angle θ=π/8 to get −cos(π/8)

4. Use the half-angle formula for cosine, which is cos(α/2)=√(,(1+cos(α))/2) for an angle in the first quadrant.

5. Set α=π/4 so that α/2=π/8

6. Evaluate the expression using cos(π/4)=√(,2)/2

7. Simplify the nested radical: cos(π/8)=√(,(1+√(,2)/2)/2)=√(,(2+√(,2))/4)=√(,2+√(,2))/2

8. Combine with the negative sign from the quadrant analysis.

Final Answer

cos((9*π)/8)=−√(,2+√(,2))/2

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