Find the Exact Value sec((7pi)/4)

Problem

sec((7*π)/4)

Solution

1. Identify the reference angle by determining how far the angle is from the nearest multiple of π

(7*π)/4=2*π−π/4

Reference Angle=π/4

2. Determine the quadrant of the angle (7*π)/4 Since 1.5*π<(7*π)/4<2*π the angle lies in Quadrant IV.

3. Recall the sign of the secant function in Quadrant IV. Since sec(θ)=1/cos(θ) and cosine is positive in Quadrant IV, secant is also positive.

sec((7*π)/4)=sec(π/4)

4. Evaluate the secant of the reference angle using the reciprocal of cos(π/4)

cos(π/4)=√(,2)/2

sec(π/4)=2/√(,2)

5. Rationalize the denominator to find the exact value.

2/√(,2)⋅√(,2)/√(,2)=(2√(,2))/2=√(,2)

Final Answer

sec((7*π)/4)=√(,2)

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