What is the value of tan 92 π 3 ?

Question

What is the value of tan 92 &#960; 3 ?

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 2 &pi; radians in a circle. Dividing 92 &pi; 3 by 2 &pi; 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 92 &pi; 3 is the result of 15 revolutions from its starting side on the positive x -axis followed by a rotation through 2 &pi; 3 radians. Therefore, the angles 92 &pi; 3 and 2 &pi; 3 are coterminal angles and tan 92 &#960; 3 is equal to tan 2 &#960; 3 . Since 2 &pi; 3 is greater than &pi; 2 and less than &pi; , it follows that the terminal side of the angle is in quadrant Iwe and forms an angle of &pi; 3 , or 60 &#176; , 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 2 &#960; 3 is 3 2 - 1 2 , which is equivalent to - 3 . Therefore, the value of tan 92 &#960; 3 is - 3 . Choice B is incorrect. This is the value of 1 tan 92 &#960; 3 , not tan 92 &#960; 3 . Choice C is incorrect. This is the value of 1 tan &#960; 3 , not tan 92 &#960; 3 . Choice D is incorrect. This is the value of tan &#960; 3 , not tan 92 &#960; 3 .

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