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

Problem

sec((7*π)/6)

Solution

1. Identify the relationship between the secant function and the cosine function using the reciprocal identity.

sec(θ)=1/cos(θ)

2. Determine the reference angle for (7*π)/6 by subtracting π from the angle, since it is in the third quadrant.

(θ_ref)=(7*π)/6−π=π/6

3. Evaluate the cosine of the reference angle using known values for special angles.

cos(π/6)=√(,3)/2

4. Determine the sign of the cosine function in the third quadrant, where both sine and cosine are negative.

cos((7*π)/6)=−√(,3)/2

5. Substitute the cosine value back into the reciprocal identity to find the secant.

sec((7*π)/6)=1/(−√(,3)/2)

6. Simplify the expression by multiplying by the reciprocal and rationalizing the denominator.

sec((7*π)/6)=−2/√(,3)⋅√(,3)/√(,3)=−(2√(,3))/3

Final Answer

sec((7*π)/6)=−(2√(,3))/3

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