Here is a unit circle with a point, , marked at . For each angle of rotation listed here, mark the new location of on the unit circle. Be prepared to explain your reasoning.

A circle with center at the origin of an x y plane. Point A lies on the outside of the circle, on the x axis, to the right of the origin. The circle is divided into 8 equal pie shaped sections.

1. ,
2. ,
3. ,
4. ,
5. ,

The center of a windmill is at and it has 5 blades, each 1 meter in length. A point, , is at the end of the blade that is pointing directly to the right of the center. Here are graphs showing the horizontal and vertical distances of point relative to the center of the windmill as the blades rotate counterclockwise.

A graph. Horizontal axis, theta, scale 0 to 5 pi, by pi over 4. Vertical axis, y, negative 1 to 1, by 0.5’s. A curve is drawn labeled y equals cosine theta.

A graph. Horizontal axis, theta, scale 0 to 5 pi, by pi over 4. Vertical axis, y, negative 1 to 1, by 0.5’s. A curve is drawn labeled y equals sine theta.

1. How many full rotations are shown by the graphs? Explain how you know.
2. What do the values of the graphs at mean in this context?
3. List some different angles of rotation that bring to the highest point in its circle of rotation. What do you notice about these angles?
4. How many angles show point at a height of 0.71 meters? Explain or show your reasoning.

1. The point  on the unit circle has coordinates . For each angle of rotation, state the number of rotations defined by the angle, and then identify the coordinates of after the given rotation.

   

   rotation in radians	number of rotations	horizontal coordinate	vertical coordinate
   	0.75	0	-1

2. In general, if is greater than  radians, explain how you can use the unit circle to make sense of and .