Find the Exact Value sec((11pi)/4)

Problem

sec((11*π)/4)

Solution

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

(11*π)/4−2*π=(11*π)/4−(8*π)/4

(11*π)/4−(8*π)/4=(3*π)/4

2. Identify the quadrant for the angle (3*π)/4 Since π/2<(3*π)/4<π the angle is in Quadrant II.

3. Determine the reference angle for (3*π)/4 in Quadrant II.

π−(3*π)/4=π/4

4. Apply the secant identity sec(θ)=1/cos(θ) In Quadrant II, the cosine (and thus secant) is negative.

sec((3*π)/4)=−sec(π/4)

5. Evaluate the exact value using the known value cos(π/4)=√(,2)/2

sec(π/4)=2/√(,2)

2/√(,2)=√(,2)

6. Combine the value and the sign determined from the quadrant.

sec((11*π)/4)=−√(,2)

Final Answer

sec((11*π)/4)=−√(,2)

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