Find the Exact Value sec(14pi)

Problem

sec(14*π)

Solution

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

2. Simplify the angle by determining how many full rotations of 2*π are in 14*π Since 14*π=7*(2*π) the angle is coterminal with 0

3. Apply the coterminal angle property to rewrite the expression.

sec(14*π)=sec(0)

4. Use the reciprocal identity for secant, which states sec(θ)=1/cos(θ)

sec(0)=1/cos(0)

5. Evaluate the cosine of 0 On the unit circle, the x-coordinate at 0 radians is 1

cos(0)=1

6. Substitute the value back into the fraction to find the final result.

1/1=1

Final Answer

sec(14*π)=1

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