Find the Exact Value sec(18pi)

Problem

sec(18*π)

Solution

1. Identify the periodicity of the secant function. The secant function, sec(θ) has a period of 2*π

2. Simplify the angle by determining the number of full rotations. Since 18*π is an even multiple of π it can be written as 9×2*π

3. Apply the periodic property sec(θ+2*π*k)=sec(θ) for any integer k In this case, sec(18*π)=sec(0+9×2*π)=sec(0)

4. Evaluate the secant function using the reciprocal identity sec(θ)=1/cos(θ)

5. Substitute the known value cos(0)=1 into the expression.

6. Calculate the final result: 1/1=1

Final Answer

sec(18*π)=1

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