Wrapping a Line Around a Circle

Topic:
Circle
THE UNIT CIRCLE: C is the circle of radius =1, centered at the origin. As we move up line L, we are going to imagine wrapping the line around C. The value of t becomes the length of the arc which has been wrapped around C. Since this is a circle of radius 1
  • wrapping around the whole circle () would require a length of .
  • wrapping around half the circle () would require a length of .
You can zoom in and out to see higher values of arc length and degrees. Every point on the line L corresponds to a point on the circle. This is called the wrapping function.

Investigations of the Wrapping Function

1. Click on the degrees, and click off the Arc Length and Right Triangle. 2. Find the x and y coordinates for the given degree values of t, listed below. Make a table and create a column with the x and y coordinates in ordered pair form:
a.t = 0o
b.t = 30o
c.t = 45o
d.t = 60o
e.t = 90o
f t= 180o 
g. t = 270o
h.t = 360o
i.t = 450o
j.t = -60o
kt = -90o
l.t = -180o
3. When you are done with that task, Click on the Right Triangle. 4. Extend your table by adding two more columns to the right of the (x,y)-column. Use the right triangle to find the sin(), cos() where is the angle at the origin shown in the right triangle. Use the t values that are already in the table for . For example, use the row corresponding to t = 30o for . Again, you don't have to make a new table, just add two columns to the existing table that you made. (If you are having trouble seeing the numbers in the right triangle, click Zoom in Right Triangle. You should PLUG THE VALUES INTO A CALCULATOR BEFORE ADDING THEM TO THE TABLE TO MAKE SURE YOU ARE ADDING THE CORRECT VALUES. 5. What is the relationship between sin() and the y-coordinate in your table? Using vocabulary from the right triangle unit, explain why? 6. What is the relationship between cos() and the x-coordinate in your table? Using vocabulary from the right triangle unit, explain why? Click on Arc Length, and Click off Right Triangle. 7. What is the arc length corresponding to the same degrees as listed above? Add a new column to your table and label it arc length. List them in terms of . These are called RADIANS. Write that above your Arc Length column. DEFINITION: A radian is the arc length in the unit circle corresponding to a given angle.