Find the Exact Value sec((13pi)/4)

Problem

sec((13*π)/4)

Solution

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

(13*π)/4−2*π=(13*π)/4−(8*π)/4

(13*π)/4−(8*π)/4=(5*π)/4

2. Identify the quadrant for the angle (5*π)/4 Since π<(5*π)/4<(3*π)/2 the angle is in Quadrant III.

3. Determine the reference angle θ′ for an angle in Quadrant III using the formula θ′=θ−π

θ′=(5*π)/4−π

θ′=π/4

4. Apply the secant definition and consider the sign in Quadrant III. In Quadrant III, the secant function is negative.

sec((5*π)/4)=−sec(π/4)

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

sec((5*π)/4)=−√(,2)

Final Answer

sec((13*π)/4)=−√(,2)

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