What is the value of tan( 92π ⁄ 3 ) ?

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Accepted Answer

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 2π ⁄ 3 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 −√3. Choice B is incorrect. This is the value of 1 ⁄ tan( 92π ⁄ 3 ) , not tan( 92π ⁄ 3 ). Choice C is incorrect. This is the value of − 1 ⁄ tan( 2π ⁄ 3 ) , not tan( 92π ⁄ 3 ). Choice D is incorrect. This is the value of tan( π ⁄ 3 ), not tan( 92π ⁄ 3 ).

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