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

Problem

sec(−(4*π)/3)

Solution

1. Apply the even-odd identity for the secant function, which states sec(−θ)=sec(θ)

sec(−(4*π)/3)=sec((4*π)/3)

2. Determine the reference angle for (4*π)/3 Since the angle is in the third quadrant, the reference angle is θ′=(4*π)/3−π

θ′=π/3

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

sec((4*π)/3)=−sec(π/3)

4. Evaluate the secant of the reference angle. Since cos(π/3)=1/2 the reciprocal is 2

sec(π/3)=2

5. Combine the sign and the value to find the final result.

−2

Final Answer

sec(−(4*π)/3)=−2

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