OG详解-OG5 数学1 Q19

Choice A is correct. A trigonometric ratio can be found using the unit circle, that is, a circle with radius 1 unit. If a central angle of a unit circle in the xy-plane centered at the origin has its starting side on the positive x-axis and its terminal side intersects the circle at a point (x, y), then the value of the tangent of the central angle is equal to the y-coordinate divided by the x-coordinate. There are 2π radians in a circle. Dividing 92π​3​ by 2π​ yields  926​​, which is equivalent to 15 + ​23​ . It follows that on the unit circle centered at the origin in the xy-plane, the angle 9​2π3​is the result of 15 revolutions from its starting side on the positive x-axis followed by a rotation through 2π​3​ radians. Therefore, the angles 9​2π3 and 2π​3are coterminal angles and tan​​(92π3​)is equal to tan(​2π3​) . Since​2π3 is greater than π​2​and less than π, it follows that the terminal side of the angle is in quadrant II and forms an angle of π3​​ , or 60°, with the negative x-axis. Therefore, the terminal side of the angle intersects the unit circle at the point (−12​,√3​2​). It follows that the value of tan(2π​3​) is √32​−​12​​   , which is equivalent to -√3​. Therefore, the value of tan(92π3​)is -√3​ .
Choice B is incorrect. This is the value of 1tan(92π​3​)​ , not tan(92π3​). Choice C is  incorrect. This is the value of 1tan(π​3​)​ , not tan(92π3​). Choice D is incorrect. This is the  value of tan(​π3​) , not tan(92​π3​) .