This section develops sine, cosine, and tangent as trigonometric functions. This builds on work from an earlier course in which students worked with sine, cosine, and tangent ratios using right triangles, and then the work in a previous section in which students used expressions involving sine and cosine to locate points on a circle.

First, students examine graphs of functions that involve repeated behavior, leading to a definition for periodic functions.

Next, students begin to plot points for the sine and cosine functions, beginning with inputs of from 0 to radians. Students then extend this work to values greater radians, both in context and abstractly.

Then, students consider inputs of negative radian values, with from 0 to radians. They make sense of these negative inputs contextually, then create graphs of and across their now extended domains.

Finally, students make sense of the tangent function, using their understanding of trigonometric ratios, the unit circle, and their work with the sine and cosine functions.

The last lesson in this section is an optional exploration of the secant, cosecant, and cotangent trigonometric functions.

This section focuses on transformations of trigonometric functions. 
Students have seen the effects of translations and scale factors on graphs in an earlier unit when they studied general transformations of functions and in a previous course when they studied quadratic equations and their graphs.

First, students are introduced to the amplitude of a periodic function in context, building from prior work with scale factor or vertical stretch of the graph of a function. Students also examine the new feature of the midline of a periodic function. 

They continue this work to combine changing amplitude and midline in the same periodic function as they relate to vertical stretch and vertical translations. Students also consider the effects of a horizontal translation on the equation and graph of the function.

Next, students examine how a horizontal scale factor affects the period of the function. Over two lessons, they put all of these transformations together to identify and determine parameters and features of trigonometric functions, including midline, amplitude, and period, using graphs and equations.

Students generalize these transformations as parameters of a periodic function, building on their work in earlier units and courses. They end the section with modeling circular motion using periodic functions. 

This section introduces students to the unit circle as it builds on work with similar triangles and trigonometric ratios. First, students consider a context with repeating outputs. Throughout the section, students use contexts involving circular motion as they make sense of periodic functions and the unit circle.

Next, students use the Pythagorean Theorem to connect legs of a right triangle to the coordinates of points on a circle. 

Then, students explore the unit circle over two lessons. Students build on the previous course work that led to defining the radian measure of an angle as the ratio of the arc length traveled to the radius of the circle. Students continue this work by creating a unit circle reference as they use the symmetry of the unit circle to identify radian angle measures and coordinates of points on the unit circle. 

Next, students connect the right triangle trigonometric ratios to the points on the circle, leading to the development of trigonometric identities over two lessons. 

Finally, students represent points on various circles in context, using expressions involving trigonometric ratios. This is an important transition step as students progress toward thinking about cosine and sine as functions.

In this final section, students have the opportunity to apply their thinking from throughout the unit. As this is a short section followed by an End-of-Unit Assessment, there are no section goals or checkpoint questions.