27-1: Circles on the Coordinate Plane
Objectives:
1. To define and use
the equation of a
circle
2. To complete and
use the unit circle
Assignment:
• P. 399: 1-9
• Complete Unit Circle
• Challenge Problems
Objective 1
You will be able to define and
use the equation of a circle
Warm-Up
When a solid is cut by a plane, the resulting
plane figure is called a section. A section
that is parallel to the base is a crosssection.
Warm-Up
Describe each of the following sections of
the double cone, none of which touch the
vertex:
1. Plane is parallel to the
“base”
2. Same plane as above
except tilted a few degrees
3. Plane is parallel to one of
the lateral sides
4. Plane is perpendicular to
the “base”
Conic Section
Here are the four basic conic sections,
which are formed by the intersection of a
plane and a double cone.
Plane does not pass through the vertex
Definition: Locus
A locus is a set of points that share a
common geometric property.
A circle is the locus of
coplanar points (𝑥, 𝑦) that
are equidistant from a
given point called the
center.
Exercise 1
Use the locus definition to derive the
equation of a circle.
Equation of a Circle
Use the locus definition to derive the
equation of a circle.
𝑥−ℎ
2
+ 𝑦−𝑘
2
= 𝑟2
Center
Radius
Exercise 2
1. Write and graph the
equation of circle
centered at the origin
with a radius of 5
inches.
2. Write and graph the
equation of a circle
centered at (-3, 2)
with a radius of 5
inches.
3. Describe a series of
transformations that
would map the circle
from Q1 to the circle
from Q2.
Objective 2
You will be
able to
complete and
use the unit
circle
Exercise 3
Solve each right triangle. Write your
answers in simplest radical form.
1 unit
1 unit
30
45
Radians
Radians are
another way to
measure an
angle. If you
take the radius
and wrap it
around the circle,
the angle that is
formed is one
radian.
Radians
It takes a little bit
more than 3
radians to span a
semicircle.
That “little bit more
than 3” is π.
So π radians = 180°
and 2π radians =
360°
Exercise 4
Rewrite each of the following angle
measures in terms of radians.
(180° = π rad)
1. 30°
2. 45°
3. 60°
4. 90°
Exercise 5
Write the equation of
the circle.
The Unit Circle
This tiny circle is
called the unit circle
since its radius is 1
unit. This circle may
be tiny, but it will give
us a way to find 102
exact trig values.
That’s pretty useful.
Unit Circle Activity
Math students often use
a unit circle to find the
exact trig ratios of
certain angle
measures, since most
calculators won’t
divulge that
information. In this
activity, we will
construct a unit circle.
y
(0, 1)
90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
The outer bold circle
is the unit circle.
We will eventually be
labeling each of the
points along this
bold circle with
ordered pairs.
y
(0, 1)
90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
Now look at the inner
circle. The points
along this circle will be
labeled with degree
measures. What do
you suppose these
degree measures
represent?
Finish the degree
measures.
y
(0, 1)
90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
Now look at the middle
circle. The points
along this circle will be
labeled with radian
measures.
Finish the radian
measures for the first
quadrant.
y
(0, 1)
90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
Finally, look at the outer
circle again. Let’s
concentrate on the
point that is 30° along
this circle.
y
(0, 1)
90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
Finally, look at the outer
circle again. Let’s
concentrate on the
point that is 30° along
this circle.
Realize that the ordered
pair for any point
makes a right triangle
with the x-axis.
y
(0, 1)
90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
1. What is the length of
the hypotenuse?
2. What is the length of
the short leg?
3. What is the length of
the longer leg?
4. What are the
coordinates of the
point on the circle?
y
(0, 1)

90
3 1
,
2 2

1
60
1
45
30
0
360
2
3
2
x
0
2
(1, 0)
Unit Circle Activity
How can this unit circle
be used to find the
following?
1. cos (30°)
2. sin (30°)
3. tan (30°)
y
(0, 1)

90
3 1
,
2 2

1
60
1
45
30
0
360
2
3
2
x
0
2
(1, 0)
Unit Circle Activity
Let’s look at the outer
circle again. This time
concentrate on the
point that is 45° along
this circle.
y
(0, 1)
90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
1. What is the length of
the hypotenuse?
2. What is the length of
the base leg?
3. What is the length of
the height leg?
4. What are the
coordinates of the
point on the circle?
y
(0, 1)

2
2
,
2
2

1
2
90
60
2
45
30
2
0 2
360
x
0
2
(1, 0)
Unit Circle Activity
How can this unit circle
be used to find the
following?
1. cos (45°)
2. sin (45°)
3. tan (45°)
y
(0, 1)

2
2
,
2
2

1
2
90
60
2
45
30
2
0 2
360
x
0
2
(1, 0)
Unit Circle Activity
In general, for any
point (x, y) along the
outer circle of the
unit circle:
1. cos( ) = x
2. sin ( ) = y
3. tan ( ) = y/x
y
(0, 1)
90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
Let’s look at the outer
circle one last time.
This time let’s look at
the point that is 150°
along the unit circle.
Obviously, we cannot
make a right triangle
with a 150° angle. So
how could we complete
the second quadrant?
y
(0, 1)
90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
The answer involves symmetry.
y
(0, 1)

-

-
2
2
3 1
,
2 2
,
2
2



2
2
,

2
2

3 1
,
2 2

90
60
45
30
0
360
x
0
2
(1, 0)
Unit Circle Activity
The same would apply for the 3rd and 4th
quadrants.
y
(0, 1)

-

-
2
2
3 1
,
2 2
,
2
2



2
2
,

2
2

3 1
,
2 2

90
60
45
30
0
360
x
0
2
(1, 0)
27-1: Circles on the Coordinate Plane
Objectives:
1. To define and use
the equation of a
circle
2. To complete and
use the unit circle
Assignment:
• P. 399: 1-9
• Complete Unit Circle
• Challenge Problems