Find the Exact Value sec((5pi)/6)

Problem

sec((5*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 (5*π)/6 by subtracting it from π

Reference Angle=π−(5*π)/6=π/6

3. Evaluate the cosine of the reference angle using the unit circle or special triangles.

cos(π/6)=√(,3)/2

4. Determine the sign of the cosine function in the second quadrant, where (5*π)/6 is located.

cos((5*π)/6)=−√(,3)/2

5. Calculate the secant value by taking the reciprocal of the cosine value.

sec((5*π)/6)=1/(−√(,3)/2)=−2/√(,3)

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

−2/√(,3)⋅√(,3)/√(,3)=−(2√(,3))/3

Final Answer

sec((5*π)/6)=−(2√(,3))/3

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