Find the Exact Value sec(8pi)

Problem

sec(8*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 removing multiples of the period. Since 8*π is an even multiple of π we can write it as 4×2*π

3. Determine the coterminal angle. The angle 8*π is coterminal with 0 on the unit circle.

sec(8*π)=sec(0)

4. Apply the reciprocal identity. The secant function is the reciprocal of the cosine function.

sec(0)=1/cos(0)

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

cos(0)=1

6. Calculate the final value by substituting the cosine value into the reciprocal expression.

1/1=1

Final Answer

sec(8*π)=1

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