Wrapping Function. This is a new function that results from wrapping a number line around a circle. P(t) = (x, y) is the wrapping function equation where t is the measure of the angle in radians (in other words, it is theta), and (x, y) is the is the ordered pair on the terminal side which intersects the circle. Recall that s = length of the arc.
1. Set theta to 0 and the circle radius to 1. This is the unit circle. Notice the number line vertically positioned tangent to the circle at (1, 0).
2. Use the slider to wrap the number line around the unit circle. Notice that the central angle = length of arc = measure of arc. Symbolically theta = s = t. This is true because r = 3. If s = r (theta), then s = (1)(theta) so s = theta = t.
4. As you wrap, continue to observe the relationship between s = t. Do they continue to be equal?
5. You can find the angle measure by creating a line segment from the center to (x, y). Observe the relationship between the angle in radians and t.
6. Lengthen the radius to r = 3. Wrap the number line around the circle until s = 3. Notice that theta is 1 radian. The relationships s = r(theta) and s/r = theta continue to hold true because it is a proportional relationship.
7. BIG IDEA 1: s = length of the arc, t = measure of the arc, theta = measure of angle in radians
8. BIG IDEA 2: s = t = theta for the unit circle because r =1
9. BIG IDEA 3: s/r = t = theta for any circle because relationship is proportional
10. BIG IDEA 4: P(t) = P(theta) = (x, y) an ordered pair on the terminal side of the angle.