Notes & FAQs

This set of standards calls for students to expand their understanding of the trigonometric functions first developed in Geometry. At first, the trigonometric functions apply only to angles in right triangles; sin θ, cos θ, and tan θ make sense only for 0<θ,pi/2 . By representing right triangles with hypotenuse 1 in the first quadrant of the plane, it can be seen that,  (cos θ, sin θ ) represents a point on the unit circle.  This leads to a natural way to extend these functions to any value of θ that remains consistent with the values for acute angles: interpreting θ as the radian measure of an angle traversed from the point (1,0) counterclockwise around the unit circle, cos θ is taken to be the x-coordinate of the point corresponding to this rotation and sin θ to be the y-coordinate of this point. This interpretation of sine and cosine immediately yields the Pythagorean Identity: that sin2(θ) + cos2(θ) = 1. This basic identity yields others through algebraic manipulation and allows values of other trigonometric functions to be found for a given θ if one of the values is known (F-TF.1, 2, 8). 

The graphs of the trigonometric functions should be explored with attention to the connection between the unit-circle representation of the trigonometric functions and their properties—for example, to illustrate the periodicity of the functions, the relationship between the maximums and minimums of the sine and cosine graphs, zeros, and so forth. Standard F-TF.5 calls for students to use trigonometric functions to model periodic phenomena. This is connected to standard F-BF.3 (families of functions), and students begin to understand the relationship between the parameters appearing in the general cosine function (and sine function) and the graph and behavior of the function (e.g., amplitude, frequency, line of symmetry).