Find the Exact Value tan((19pi)/6)

Problem

tan((19*π)/6)

Solution

1. Subtract multiples of the period to find the coterminal angle within the standard range. Since the period of the tangent function is π we can subtract 3*π (which is (18*π)/6 from the angle.

(19*π)/6−(18*π)/6=π/6

2. Apply the periodicity identity tan(θ+n*π)=tan(θ)

tan((19*π)/6)=tan(π/6)

3. Evaluate the tangent of the reference angle π/6 (or 30.

tan(π/6)=sin(π/6)/cos(π/6)

4. Substitute the known values for sine and cosine at π/6

tan(π/6)=(1/2)/(√(,3)/2)

5. Simplify the fraction by canceling the denominators.

tan(π/6)=1/√(,3)

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

1/√(,3)⋅√(,3)/√(,3)=√(,3)/3

Final Answer

tan((19*π)/6)=√(,3)/3

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