Find the Exact Value sec((15pi)/8)

Problem

sec((15*π)/8)

Solution

1. Identify the reference angle by finding the distance to the nearest multiple of π Since (15*π)/8 is in the fourth quadrant, the reference angle is 2*π−(15*π)/8=π/8

2. Determine the sign of the function. In the fourth quadrant, the cosine function is positive, which means its reciprocal, the secant function, is also positive.

3. Apply the reciprocal identity to relate the expression to the cosine function.

sec((15*π)/8)=1/cos(π/8)

4. Use the half-angle formula for cosine, cos(θ/2)=√(,(1+cos(θ))/2) where θ=π/4

cos(π/8)=√(,(1+cos(π/4))/2)

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

cos(π/8)=√(,(1+√(,2)/2)/2)

6. Simplify the expression inside the square root.

cos(π/8)=√(,(2+√(,2))/4)

cos(π/8)=√(,2+√(,2))/2

7. Calculate the secant by taking the reciprocal of the cosine value.

sec((15*π)/8)=2/√(,2+√(,2))

8. Rationalize the denominator by multiplying the numerator and denominator by √(,2−√(,2))

sec((15*π)/8)=(2√(,2−√(,2)))/√(,(2+√(,2))*(2−√(,2)))

sec((15*π)/8)=(2√(,2−√(,2)))/√(,4−2)

sec((15*π)/8)=(2√(,2−√(,2)))/√(,2)

9. Simplify the final radical expression.

sec((15*π)/8)=√(,2)⋅√(,2−√(,2))

sec((15*π)/8)=√(,4−2√(,2))

Final Answer

sec((15*π)/8)=√(,4−2√(,2))

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