Find the Exact Value sec(-(7pi)/6)

Problem

sec(−(7*π)/6)

Solution

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

sec(−(7*π)/6)=sec((7*π)/6)

2. Identify the reference angle by determining the position of (7*π)/6 on the unit circle.

(7*π)/6=π+π/6

3. Determine the quadrant and the sign of the secant function. Since (7*π)/6 is in Quadrant III, the cosine (and thus secant) is negative.

sec((7*π)/6)=−sec(π/6)

4. Evaluate the secant of the reference angle π/6 using the reciprocal identity sec(θ)=1/cos(θ)

cos(π/6)=√(,3)/2

5. Calculate the reciprocal to find the value of the secant.

sec(π/6)=2/√(,3)

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

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

7. Combine the results by applying the negative sign from the quadrant analysis.

−(2√(,3))/3

Final Answer

sec(−(7*π)/6)=−(2√(,3))/3

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