Find the Exact Value sec((8pi)/3)

Problem

sec((8*π)/3)

Solution

1. Find the coterminal angle by subtracting multiples of 2*π until the angle is within the interval [0,2*π)

(8*π)/3−2*π=(8*π)/3−(6*π)/3

(8*π)/3−(6*π)/3=(2*π)/3

2. Identify the reference angle for (2*π)/3 in the second quadrant.

π−(2*π)/3=π/3

3. Determine the sign of the secant function in the second quadrant. Since cos(θ) is negative in the second quadrant, sec(θ) is also negative.

sec((2*π)/3)=−sec(π/3)

4. Evaluate the secant using the reciprocal of the cosine value for the reference angle.

cos(π/3)=1/2

sec(π/3)=1/(1/2)=2

5. Apply the sign to find the final value.

sec((2*π)/3)=−2

Final Answer

sec((8*π)/3)=−2

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