Find the Exact Value sec((23pi)/6)

Problem

sec((23*π)/6)

Solution

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

(23*π)/6−2*π=(23*π)/6−(12*π)/6

(11*π)/6

2. Identify the reference angle for (11*π)/6 which is located in the fourth quadrant.

2*π−(11*π)/6=π/6

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

sec((11*π)/6)=sec(π/6)

4. Apply the reciprocal identity sec(θ)=1/cos(θ)

sec(π/6)=1/cos(π/6)

5. Substitute the known value cos(π/6)=√(,3)/2

1/√(,3)/2=2/√(,3)

6. Rationalize the denominator by multiplying the numerator and denominator by √(,3)

(2√(,3))/3

Final Answer

sec((23*π)/6)=(2√(,3))/3

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