Math 142 Pre-Calc II Notes
5.1 The Unit Circle
Objectives: Find coordinates of terminal points on the unit circle; use reference numbers.
The Unit Circle is a circle centered at the origin with radius 1. The equation of the unit circle is:
x2 + y 2 = 1
6
-
Show that the point
Example 2
If  , y is a point in quadrant IV on the unit circle. Find y.
3
,
3
Example 1
3
is a point on the unit circle.
1
2
Terminal Points on the Unit Circle
Suppose t is a real number. If we mark off a length of t on the unit circle we get a terminal point on the circle.
P(x, y)
t>0
1
1
t<0
P(x, y)
Example 3
(a) t =
Find the terminal point for each real number:
π
2
(b) t = π
(c) t = -3 π
(d) t =
π
4
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2
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Reference Numbers
A reference number, t, for the number t is the shortest distance along the unit circle to the x-axis regardless of direction.
t
t
t
t
1
1
Two triangles that will give additional “nice” exact values are a 45 °- 45°- 90° triangle, t =
gle, t =
π
and t =
6
π
3
π
4
, and a 30°- 60°- 90° trian-
:
30°- 60°- 90°
45°- 45°- 90°
y
y
π
4
45°
2
1
1 ⟹
π
2
1
1
2
1
45°
x
1
⟹
2
3
2
3
Find the coordinates for each terminal point on the unit circle:
Example 4
π
2
π
3
π
4
π
6
0
Find the coordinates of the terminal point for:
Example 5
(a)
t=-
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5π
4
(b)
t=
5π
6
1
30°
2
1
6
60°
(c)
t=
-π
3
(d)
11 π
2
x