If students are unsure where to start graphing, consider asking:

• “How could you use the unit circle or technology to help you?”

• “Consider the angles , , , and . How could you use the value of cosine at each of these angles to make the graph?

Display completed graphs for cosine and sine from the activity, alongside a unit circle for reference, throughout the discussion. Begin the discussion by selecting previously identified students to share their observations. If no one made the connection to the Warm-up descriptions and the current graphs, invite students to do so now, highlighting how the graph of cosine matches the description for the -coordinates and the graph of sine matches the description for the -coordinates.

Next, invite students to share whether they think these graphs represent functions. To help convince students that they are functions, ask, “What are the inputs? The outputs?” (The input is an angle in radians on the unit circle. The output is either the - or -coordinate of the point on the unit circle associated with the angle.) Since each angle has only one output associated with it, which we can visualize by drawing a line using the angle from the origin intersecting the unit circle at a single point, cosine and sine are functions.

Some students may think   or   has special meaning and may be unsure how to make a prediction about their graphs. Ask these students how they would write the expressions on the left without using the square (  and  ).

Select previously identified students to share how they identified  and  as the two inputs that make   true. If possible, display any student sketches of the unit circle.

Invite students to share how their predictions matched up to their partner’s and the actual graphs. Some important takeaways from this discussion are:

• The cosine and sine functions are just like other functions we have studied. You can graph them and combine them with other functions using basic operations, like addition and multiplication.
• If you square cosine or sine, the output values are all positive or 0.
• The graph of   looks just like the graph of  and is a visual of the Pythagorean Identity, which states that   is always true.

During the graphing process, students may take note that the graph shows values for cosine and sine beyond inputs from 0 to . If they question how this is possible and time allows, invite students to suggest the meaning of these inputs. Thinking about inputs outside of 0 to is the focus of the next lesson.