Find the Exact Value sec((15pi)/4)

Problem

sec((15*π)/4)

Solution

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

(15*π)/4−2*π=(15*π)/4−(8*π)/4

(7*π)/4

2. Identify the quadrant for the angle (7*π)/4 Since (3*π)/2<(7*π)/4<2*π the angle is in Quadrant IV.

3. Determine the reference angle θ′ for an angle in Quadrant IV using the formula θ′=2*π−θ

θ′=2*π−(7*π)/4

θ′=π/4

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

sec((7*π)/4)=sec(π/4)

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

sec(π/4)=1/√(,2)/2

sec(π/4)=2/√(,2)

sec(π/4)=√(,2)

Final Answer

sec((15*π)/4)=√(,2)

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