Display the unit circle with points marked. If students need extra practice working with radians, select students to share the strategies they used to place these points, such as by sectioning off the unit circle into equal-sized pieces.

Invite students to share their thinking for the locations of points and . A key point here is that angle measurements can go beyond radians. Here are some questions for discussion:

• “What do you think the cosine value at radians is? At radians?” (The value of cosine at these angles is 0 and -1 since these angles correspond to the same point on the unit circle as  and radians, respectively.)
• “What do you think the sine of radians is? radians? radians?” (The value of sine at all of these angles is 0, because they are all different numbers of full rotations and sine has a value of 0 at 0 radians.)

If the students cannot yet separate circular motion into vertical change and horizontal change, consider asking:

• “What can you tell me about how moves horizontally (left and right) as it goes around the windmill?”

• “Sketch a unit circle. Place your finger on the circle at 0 radians, and slowly trace around the circle counterclockwise. What do you notice about how your finger moves horizontally (left and right)?”

Display the two graphs from the activity, and refer to them throughout the discussion. Begin the discussion by inviting students to state angles that bring to its highest point in its circle of rotation. While students don’t need to generalize to an expression like , they should understand that adding any multiple of to will give an angle for which the blade with point  is pointing directly up.

Next, select 2–3 students to share their reasoning for the number of angles that show at a height of 0.71 meters. If students do not suggest graphing the line to see where it intersects the graph of , do so, and ask students to interpret what the points of intersection mean.

Begin the discussion by selecting students who determined the angle between 0 and that matched the given angle in the table and then used the values from a unit circle display, and ask them to share their reasoning. Then invite students who used technology to determine the coordinates by calculating and . Both methods work, but the latter takes advantage of cosine and sine as functions whose domain extends beyond radians.

Highlight the values at and  radians. Both of these angles describe an arc starting and stopping at the same points on the unit circle, but the second completes one extra full rotation before stopping. Remind students that while some values of cosine and sine appear more often than every  radians, the entire set of values for each repeats every  radians, and this is why the period of these functions is .