Find the Exact Value sec(4pi)

Problem

sec(4*π)

Solution

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

2. Simplify the angle by subtracting multiples of the period. Since 4*π is an even multiple of 2*π the value of sec(4*π) is equivalent to the value of sec(0)

sec(4*π)=sec(4*π−2*(2*π))

sec(4*π)=sec(0)

3. Apply the reciprocal identity for secant. The secant function is defined as the reciprocal of the cosine function.

sec(0)=1/cos(0)

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

cos(0)=1

5. Calculate the final result by substituting the value of the cosine back into the reciprocal expression.

sec(0)=1/1

sec(0)=1

Final Answer

sec(4*π)=1

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