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Difficulty: Hard

What is the value of $tangent StartFraction 92 pi Over 3 EndFraction$ ?

$minus StartRoot 3 EndRoot$

$- \frac{\sqrt{3}}{3}$

$StartFraction StartRoot 3 EndRoot Over 3 EndFraction$

$StartRoot 3 EndRoot$

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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 $2 pi$ radians in a circle. Dividing $StartFraction 92 pi Over 3 EndFraction$ by $2 pi$ yields $StartFraction 92 Over 6 EndFraction$ , which is equivalent to $15 plus two thirds$ . It follows that on the unit circle centered at the origin in the xy-plane, the angle $StartFraction 92 pi Over 3 EndFraction$ is the result of $15$ revolutions from its starting side on the positive x-axis followed by a rotation through $StartFraction 2 pi Over 3 EndFraction$ radians. Therefore, the angles $StartFraction 92 pi Over 3 EndFraction$ and $StartFraction 2 pi Over 3 EndFraction$ are coterminal angles and $tangent left parenthesis StartFraction 92 pi Over 3 EndFraction right parenthesis$ is equal to $tangent left parenthesis StartFraction 2 pi Over 3 EndFraction right parenthesis$ . Since $StartFraction 2 pi Over 3 EndFraction$ is greater than $StartFraction pi Over 2 EndFraction$ and less than $π$ , it follows that the terminal side of the angle is in quadrant II and forms an angle of $StartFraction pi Over 3 EndFraction$ , or $60 degree$ , with the negative x-axis. Therefore, the terminal side of the angle intersects the unit circle at the point $(- \frac{1}{2}, \frac{\sqrt{3}}{2})$ . It follows that the value of $tangent left parenthesis StartFraction 2 pi Over 3 EndFraction right parenthesis$ is $\frac{\frac{\sqrt{3}}{2}}{- \frac{1}{2}}$ , which is equivalent to $minus StartRoot 3 EndRoot$ . Therefore, the value of $tangent left parenthesis StartFraction 92 pi Over 3 EndFraction right parenthesis$ is $minus StartRoot 3 EndRoot$ .

Choice B is incorrect. This is the value of $StartStartFraction 1 OverOver tangent left parenthesis StartFraction 92 pi Over 3 EndFraction right parenthesis EndEndFraction$ , not $tangent left parenthesis StartFraction 92 pi Over 3 EndFraction right parenthesis$ .

Choice C is incorrect. This is the value of $StartStartFraction 1 OverOver tangent left parenthesis StartFraction pi Over 3 EndFraction right parenthesis EndEndFraction$ , not $tangent left parenthesis StartFraction 92 pi Over 3 EndFraction right parenthesis$ .

Choice D is incorrect. This is the value of $tangent left parenthesis StartFraction pi Over 3 EndFraction right parenthesis$ , not $tangent left parenthesis StartFraction 92 pi Over 3 EndFraction right parenthesis$ .