Final answer:
The exact value of tan 11π/6 is found by realizing it's equivalent to tan(-π/6) or tan(30°), which is -√3/3.
Step-by-step explanation:
The question asks to find the exact value of the trigonometric function tan 11π/6. To solve this, we need to recognize that 11π/6 is an angle that is in standard position (its vertex is at the origin and its initial side is on the positive x-axis) and has been rotated clockwise, which is equivalent to -π/6 radians counter-clockwise. This angle corresponds to a 30-degree angle (or π/6 radians) in the fourth quadrant where the tangent is negative.
Since the tangent function has a period of π, tan(11π/6) = tan(11π/6 - 2π) = tan(-π/6), which is the same as tan(30°). The exact value of tan(30°) is -√3/3 because in a 30-60-90 right triangle, the opposite side to the 30° angle is half the hypotenuse, and the adjacent side is the hypotenuse times √3/2. Therefore, the tangent (opposite/adjacent) of 30° is 1/(√3/2) = 2/√3, which simplifies to √3/3 when rationalized.