Find the Exact Value sec((3pi)/8)

Problem

sec((3*π)/8)

Solution

1. Identify the angle as half of a known angle from the unit circle. Since (3*π)/8=1/2⋅(3*π)/4 we can use the half-angle identity for cosine.

2. Recall the half-angle identity for cosine, which is cos(θ/2)=±√(,(1+cos(θ))/2)

3. Determine the sign of the result. Since (3*π)/8 is in the first quadrant (0<(3*π)/8<π/2, the cosine value is positive.

4. Substitute θ=(3*π)/4 into the identity.

cos((3*π)/8)=√(,(1+cos((3*π)/4))/2)

5. Evaluate cos((3*π)/4) which is −√(,2)/2

cos((3*π)/8)=√(,(1−√(,2)/2)/2)

6. Simplify the expression inside the radical by finding a common denominator.

cos((3*π)/8)=√(,(2−√(,2))/2/2)

cos((3*π)/8)=√(,(2−√(,2))/4)

cos((3*π)/8)=√(,2−√(,2))/2

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

sec((3*π)/8)=1/cos((3*π)/8)

sec((3*π)/8)=2/√(,2−√(,2))

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

sec((3*π)/8)=(2√(,2+√(,2)))/√(,(2−√(,2))*(2+√(,2)))

sec((3*π)/8)=(2√(,2+√(,2)))/√(,4−2)

sec((3*π)/8)=(2√(,2+√(,2)))/√(,2)

9. Simplify the fraction by dividing by √(,2)

sec((3*π)/8)=√(,2)⋅√(,2+√(,2))

sec((3*π)/8)=√(,4+2√(,2))

Final Answer

sec((3*π)/8)=√(,4+2√(,2))

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