Find the Exact Value cos((27pi)/8)

Problem

cos((27*π)/8)

Solution

1. Find the coterminal angle within the interval [0,2*π) by subtracting multiples of 2*π

(27*π)/8−2*π=(27*π)/8−(16*π)/8

(11*π)/8

2. Identify the quadrant for the angle (11*π)/8 Since π<(11*π)/8<(3*π)/2 the angle is in Quadrant III, where cosine is negative.

3. Use the half-angle formula for cosine, cos(θ/2)=±√(,(1+cos(θ))/2) letting θ/2=(11*π)/8 which means θ=(11*π)/4

4. Find the cosine of the doubled angle (11*π)/4 This is coterminal with (3*π)/4 because (11*π)/4−2*π=(3*π)/4

cos((11*π)/4)=cos((3*π)/4)

cos((3*π)/4)=−√(,2)/2

5. Substitute the value into the half-angle formula, choosing the negative sign because the original angle is in Quadrant III.

cos((11*π)/8)=−√(,(1+(−√(,2)/2))/2)

6. Simplify the expression inside the radical.

cos((11*π)/8)=−√(,(2−√(,2))/2/2)

cos((11*π)/8)=−√(,(2−√(,2))/4)

cos((11*π)/8)=−√(,2−√(,2))/2

Final Answer

cos((27*π)/8)=−√(,2−√(,2))/2

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