Find the Exact Value sec((5pi)/4)

Problem

sec((5*π)/4)

Solution

1. Identify the reference angle by determining how far the angle is from the nearest xaxis. Since (5*π)/4 is in the third quadrant, the reference angle is (5*π)/4−π=π/4

2. Determine the sign of the secant function in the third quadrant. In the third quadrant, only tangent and cotangent are positive, so sec((5*π)/4) must be negative.

3. Recall the reciprocal identity for secant.

sec(θ)=1/cos(θ)

4. Evaluate the cosine of the reference angle.

cos(π/4)=√(,2)/2

5. Apply the sign and the reciprocal to find the final value.

sec((5*π)/4)=−1/cos(π/4)

sec((5*π)/4)=−2/√(,2)

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

−(2√(,2))/2=−√(,2)

Final Answer

sec((5*π)/4)=−√(,2)

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