Unit 6, Lesson 10
Beyond $2\pi$
Integrated Math 3
Lesson Synthesis
Periodic functions repeat their values at regular intervals. The goal of this discussion is to show how and why the values of the cosine function repeat. Display this unit circle:
Tell students that point is the result of rotating point counterclockwise by radians. It is also the result of rotating point counterclockwise by radians. Display this graph of :
Ask students,
- “How is the point at related to point ?” (This point is the horizontal position of as it moves around the unit circle, or the cosine of the angle as travels around the unit circle. The cosine of is 0, and the horizontal position of point is 0 after having been rotated radians from .)
- “How is the point at related to point ?” (Since is in the same position on the unit circle after rotating radians and rotating radians, its horizontal position is still 0. That is, must equal , which is 0.)
- “Which point on the unit circle corresponds to ?” (This point is associated with on the unit circle, which is where the value of the -coordinate goes from negative to positive as increases. While the -coordinate is 0, the -coordinate is not the same as when the value of the -coordinate goes from positive to negative as increases.)
- “How can you tell that for any angle ?” (Every time you add to the angle, the position of the point is the same on the unit circle. Since the horizontal position of the point is the same, the cosine of the angle is also the same.)
Student Lesson Summary
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.