Find the Exact Value cos(9/8*pi)

Problem

cos((9*π)/8)

Solution

1. Identify the angle and its quadrant. The angle (9*π)/8 is in the third quadrant because π<(9*π)/8<(3*π)/2 In the third quadrant, the cosine function is negative.

2. Determine the reference angle. The reference angle is found by subtracting π from the given angle.

(θ_ref)=(9*π)/8−π=π/8

3. Apply the half-angle formula for cosine. The formula is cos(θ/2)=±√(,(1+cos(θ))/2) Let θ/2=π/8 which means θ=π/4

4. Substitute the value of cos(π/4)=√(,2)/2 into the formula.

cos(π/8)=√(,(1+√(,2)/2)/2)

5. Simplify the expression inside the radical.

cos(π/8)=√(,(2+√(,2))/2/2)=√(,2+√(,2))/2

6. Assign the correct sign based on the quadrant of the original angle. Since (9*π)/8 is in the third quadrant, the value must be negative.

cos((9*π)/8)=−√(,2+√(,2))/2

Final Answer

cos((9*π)/8)=−√(,2+√(,2))/2

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