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

Problem

tan((13*π)/6)

Solution

1. Find the coterminal angle by subtracting 2*π from the given angle to find an equivalent angle within the standard interval [0,2*π)

(13*π)/6−2*π=(13*π)/6−(12*π)/6

(13*π)/6−(12*π)/6=π/6

2. Identify the trigonometric identity for the tangent of the coterminal angle.

tan((13*π)/6)=tan(π/6)

3. Evaluate the tangent using the known values for the special angle π/6 (or 30.

tan(π/6)=sin(π/6)/cos(π/6)

tan(π/6)=(1/2)/(√(,3)/2)

4. Simplify the fraction by canceling the denominators and rationalizing if necessary.

tan(π/6)=1/√(,3)

1/√(,3)⋅√(,3)/√(,3)=√(,3)/3

Final Answer

tan((13*π)/6)=√(,3)/3

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