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Type: Multiple-Choice
Category: Trigonometry
Level: Grade 11
Standards: HSF-TF.A.4
Author: nsharp1
Created: 4 years ago

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Trigonometry Question

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Let circle A be the unit circle centered at the origin. Assume that x1,x2,x3,x4,y1,y2,y3,y4 are all real numbers that are greater than or equal to zero.

Grade 11 Trigonometry CCSS: HSF-TF.A.4

Using the same reasoning as was done with the sine function, it can be shown that the tangent function is odd. However, there are values for which the tangent function is undefined (its domain is not all real numbers). How is this possible, given that the unit circle has no undefined values on its circumference?
  1. Since the tangent function has a period of π instead of 2π, it is actually associated with the unit semicircle centered at the origin. Therefore, because of the discontinuity of it being a semicircle instead of a full circle, the undefined values arise.
  2. The values of the tangent function do not occur on circle A. Referring to a right triangle, the tangent of either acute angle is the opposite side divided by the adjacent side. Since this value does not include the hypotenuse, the points on circle A are actually only an approximation of the tangent function values. Therefore, although the tangent function has undefined values, these do not arise in the approximated values of circle A.
  3. Since the values of tan(Θ) can be represented as Y1X1, the values where X1=0 are undefined and correspond to the domain restrictions of the tangent function.
  4. Since the values of tan(Θ) can be represented as X1Y1, the values where Y1=0 are undefined and correspond to the domain restrictions of the tangent function.