Find the Exact Value cos((17pi)/8)

Problem

cos((17*π)/8)

Solution

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

(17*π)/8−2*π=(17*π)/8−(16*π)/8

(17*π)/8−(16*π)/8=π/8

2. Apply the cosine half-angle identity cos(θ/2)=±√(,(1+cos(θ))/2) where θ=π/4

cos(π/8)=√(,(1+cos(π/4))/2)

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

cos(π/8)=√(,(1+√(,2)/2)/2)

4. Simplify the expression by finding a common denominator in the numerator and dividing by 2.

cos(π/8)=√(,(2+√(,2))/2/2)

cos(π/8)=√(,(2+√(,2))/4)

5. Simplify the radical by taking the square root of the denominator.

cos(π/8)=√(,2+√(,2))/2

Final Answer

cos((17*π)/8)=√(,2+√(,2))/2

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