Find the Exact Value csc(pi/8)

Problem

csc(π/8)

Solution

1. Identify the relationship between the cosecant function and the sine function using the reciprocal identity.

csc(π/8)=1/sin(π/8)

2. Apply the half-angle formula for sine, which states sin(θ/2)=√(,(1−cos(θ))/2) for an angle in the first quadrant. Let θ=π/4

sin(π/8)=√(,(1−cos(π/4))/2)

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

sin(π/8)=√(,(1−√(,2)/2)/2)

4. Simplify the expression inside the square root by finding a common denominator in the numerator.

sin(π/8)=√(,(2−√(,2))/2/2)

sin(π/8)=√(,(2−√(,2))/4)

sin(π/8)=√(,2−√(,2))/2

5. Calculate the cosecant by taking the reciprocal of the sine value found in the previous step.

csc(π/8)=2/√(,2−√(,2))

6. Rationalize the denominator by multiplying the numerator and denominator by √(,2+√(,2))

csc(π/8)=(2√(,2+√(,2)))/√(,(2−√(,2))*(2+√(,2)))

csc(π/8)=(2√(,2+√(,2)))/√(,4−2)

csc(π/8)=(2√(,2+√(,2)))/√(,2)

7. Simplify the final expression by dividing 2 by √(,2)

csc(π/8)=√(,2)√(,2+√(,2))

csc(π/8)=√(,4+2√(,2))

Final Answer

csc(π/8)=√(,4+2√(,2))

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