# Wrapping a Line Around a Circle

- Author:
- L. Marizza A. Bailey, Ron Smith

- 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 .

**wrapping function**.

**Investigations of the Wrapping Function **

**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 = 0^{o} |

b. | t = 30^{o} |

c. | t = 45^{o} |

d. | t = 60^{o} |

e. | t = 90^{o} |

f | t= 180^{o} |

g. | t = 270^{o} | |

h. | t = 360^{o} | |

i. | t = 450^{o} | |

j. | t = -60^{o} | |

k | t = -90^{o} | |

l. | t = -180^{o} | |

**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 = 30

^{o }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.