Find the Exact Value sec((13pi)/6)

Problem

sec((13*π)/6)

Solution

1. Find the coterminal angle by subtracting 2*π from the given angle to find an equivalent angle within the first rotation.

(13*π)/6−2*π=(13*π)/6−(12*π)/6

(13*π)/6−(12*π)/6=π/6

2. Apply the reciprocal identity for the secant function, which states that sec(θ)=1/cos(θ)

sec(π/6)=1/cos(π/6)

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

cos(π/6)=√(,3)/2

4. Calculate the reciprocal of the cosine value to find the secant.

sec(π/6)=1/√(,3)/2

sec(π/6)=2/√(,3)

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

2/√(,3)⋅√(,3)/√(,3)=(2√(,3))/3

Final Answer

sec((13*π)/6)=(2√(,3))/3

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