Find the Exact Value sec((4pi)/3)

Problem

sec((4*π)/3)

Solution

1. Identify the reference angle by determining how far the angle (4*π)/3 is from the nearest xaxis.

Reference Angle=(4*π)/3−π=π/3

2. Determine the quadrant of the angle (4*π)/3 Since π<(4*π)/3<(3*π)/2 the angle lies in Quadrant III.

3. Recall the sign of the secant function in Quadrant III. In Quadrant III, the cosine and secant functions are negative.

sec((4*π)/3)=−sec(π/3)

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

cos(π/3)=1/2

sec(π/3)=2

5. Apply the negative sign determined from the quadrant.

sec((4*π)/3)=−2

Final Answer

sec((4*π)/3)=−2

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