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

Problem

sec(−π/6)

Solution

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

sec(−π/6)=sec(π/6)

2. Use the reciprocal identity to express secant in terms of cosine.

sec(π/6)=1/cos(π/6)

3. Substitute the known value for cos(π/6) which is √(,3)/2

1/cos(π/6)=1/√(,3)/2

4. Simplify the fraction by multiplying by the reciprocal.

1/√(,3)/2=2/√(,3)

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

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

Final Answer

sec(−π/6)=(2√(,3))/3

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