Find the Exact Value sec(5pi)

Problem

sec(5*π)

Solution

1. Identify the periodicity of the secant function. The secant function has a period of 2*π meaning sec(θ)=sec(θ+2*k*π) for any integer k

2. Simplify the angle by subtracting multiples of 2*π Since 5*π=4*π+π the value is equivalent to sec(π)

3. Apply the definition of secant in terms of cosine. The secant function is the reciprocal of the cosine function.

sec(π)=1/cos(π)

4. Evaluate the cosine of π On the unit circle, the coordinates at π radians are (−1,0) so cos(π)=−1

5. Calculate the final reciprocal value.

1/(−1)=−1

Final Answer

sec(5*π)=−1

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