Find the Exact Value sec(-5pi)

Problem

sec(−5*π)

Solution

1. Use the property of even functions to simplify the negative angle, noting that sec(−θ)=sec(θ)

sec(−5*π)=sec(5*π)

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

5*π−2*π−2*π=π

3. Rewrite the expression using the coterminal angle.

sec(5*π)=sec(π)

4. Apply the reciprocal identity for the secant function, which is sec(x)=1/cos(x)

sec(π)=1/cos(π)

5. Evaluate the cosine of π using the unit circle, where cos(π)=−1

1/cos(π)=1/(−1)

6. Simplify the fraction to find the final value.

1/(−1)=−1

Final Answer

sec(−5*π)=−1

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