What is the value of tan 92π/3?

Explanation

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 92/6, which is equivalent to 15 + 2/3. It follows that on the unit circle centered at the origin in the xy-plane, the angle 92π/3 is the result of 15 revolutions from its starting side on the positive x-axis followed by a rotation through radians. Therefore, the angles 92π/3 and 2π/3 are coterminal angles and tan (92π/3) is equal to tan (2π/3). Since 2π/3 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 π/3, or 60°, with the negative x-axis. Therefore, the terminal side of the angle intersects the unit circle at the point (-1/2, √3/2). It follows that the value of tan(2π/3) is √3/2/-1/2 , which is equivalent to -√3. Therefore, the value of tan (92π/3) is -v3.