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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.
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.
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 | |
In general, if is greater than radians, explain how you can use the unit circle to make sense of and .
Here is a wheel with a radius of 1 foot and its center at . What happens to point if we rotate the wheel one full circle? Two full circles? 12 full circles? To someone who didn’t watch us turn the wheel, it would look like we had not moved the wheel at all since all of these rotations take point right back to where it started.
Here is a table showing different counterclockwise rotation amounts and the -coordinate of the corresponding point.
Notice that when the angle measure increases by , the -coordinate of point stays the same. This makes sense because radians is a full rotation. In terms of functions, the cosine function, , gives the -coordinate of the point on the unit circle corresponding to radians. Since one full circle is radians, this means for any input , adding multiples of will not change the value of the output. As seen in the table, the value of is the same as , which is the same as , and so on.
| rotations | angle measure in radians |
-coordinate of |
|---|---|---|
| 0 | ||
| 1 | 1 | |
| 0 | ||
| 2 | 1 | |
| 0 | ||
| 3 | 1 |
We can also see this from the graph of . To see when takes the value 1, here is a graph of and . Notice that they meet each time the input changes by , which makes sense as that represents one full rotation around the unit circle.
A graph. Horizontal axis, theta, scale 0 to 6 pi, by pi. Vertical axis, y, negative 1 to 1, by 1s. The graph of a cosine curve is drawn passing through the points 0 comma 1, pi comma negative 1, 2 pi comma 1, 3 pi comma negative 1, and so on. A horizontal line is drawn at y equals 1 and intersections the cosine curve at the top of each wave.