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5.1 The Unit Circle
Essential Question:
What is the unit circle?
What is the unit circle?
The unit circle is the circle of radius 1
centered at the origin in the xy-plane. Its
equation is
x²+y²=1
Example:
Show that the point P  3 ,
 3

2  is on the unit circle.

3
Example:
The point P (x,y) is on the unit circle. If the
y-coordinate of P is -1/3 and P is in
quadrant IV, find the x-coordinate.
Example:
The point P (x,y) is on the unit circle. Find the
missing coordinate from the given information:
2
x  coordinate: 
5
P is in quadrant II
Three points to learn:
  3 1
, 

6  2 2
  2 2
,


4  2 2 
 1
3
 ,

3 2 2 
What is a reference number?
Let t be a real number. The
reference number t’ associated with t
is the shortest distance along the unit
circle between the terminal point
determined by t and the x-axis.
Examples: Find the reference number
5
4
1. t 
2. t 
3
7
11
3. t 
6
7
4. t 
2
Using reference numbers to find
terminal points:
1. Find the reference number t’.
2. Find the terminal point determined(by
t’.
a , b )
3. The terminal point determined by t is
P (± a, ± b), where the signs are chosen
according to the quadrant in which the
terminal point lies.
Examples: Find the terminal point
determined by t
4
1. t 
3
5
3. t 
6
5
2. t 
2
7
4. t 
4