The Unit Circle
Exploring a circle that just isn’t any
circle
A FUN LEARNING PRESENTATION BY
WENDELL COOK
05 FEBRUARY 2015
What we will learn today

What is the Unit Circle?

Why is it so important?

Where did it come from?

So what do I do with it? – Angles

So what do I do with it? – Radians

So what do I do with it? – Triangle Sides

So what do I do with it? - Lengths

Closing with a song..

Additional information
What is the Unit Circle
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
It is a circle that has a radius of 1 (no
units)

Its simplicity allows demonstrating the
relationship between angles and lengths

The origin is located where the axes cross
which is in the center of the circle as
shown
Why is it so important?

The unit circle provides a foundation for the study of Geometry

It provides a demonstration of the fundamental functions of
Trigonometry

It allows a graphic display of the polar coordinate system (r,φ)
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Where did it come from?
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
Although it is not clear when the Unit Circle became in
vogue it was most likely used in the very early origins of
Geometry

Hipparchus of Nicea (somewhere near Greece) is
considered one of the Fathers of Trigonometry. He
developed a table of side ratios to help astonomers
and most likely used a unit circle.

The ancient Sumerians divided the circle into 360
portions and developed the sexigesimal number
system(base 60) since so many numbers are factors of
60. (2,3,4,5,6,10,12,15..)
So what do I do with it? - Degrees
Provides a visual of the more
important angles for study in degrees
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
The unit circle assigns the value of “0
degrees” to the horizontal axis
crossing of the circle on the right
side.

The angle value increases counterclockwise from 0 degrees

One “trip” around the circle back to
the beginning transverses 360
degrees.

As a matter of fact the circle can be
tranversed multiple times resulting in
each position having multiple values
(e.g. 0 =360=720…) as shown in the
figure
Recommended You Commit Angles
on the Circle To Memory
So what do I do with it? - Radians
Shows how those angles are
represented in radians
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
The circle can be divided in
units of radians

180 degrees is equivalent to π
(pronounced “Pi”) radians. The
value for Pi is 3.14 radians

Important angles on the Unit
Circle are shown in radians
Exercise: Create a table of the important angles in
radians shown in the figure and list their equivalent
angles in degrees (from previous chart)
So what do I do with it? – Triangle
Sides
 While the radius always
1
remains “1” the
horizontal and vertical
components vary with
angle
60
1/2
30

The Unit Circle
demonstrates this

Note how the “30-60-90
triangle is represented
on the unit circle
√3/2
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Exercise: Add the vertical and horizontal components
of the radius corresponding to the angle shown in the
table produced in the previous excercise
So what do I do with it? - Lengths

The Short Length is ½

The Medium Length is √2/2

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The Long Length is √3/2
Recommended You Commit This To Memory
Closing with a happy song..

If you are going to study trigonometry
the Unit Circle is your friend.

Learn the relationships between
lengths and angles

Be comfortable with converting
degrees to radians (180 degrees = π
radians)
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Additional Information
Clink these links
to get additional
information!!
(Wish I had
internet 2000
years ago)
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Unit Circle Link 1
Unit Circle Link 4
Unit Circle Link 2
Unit Circle Link 5
Unit Circle Link 3