Unwrapping the Unit Circle

1
Unwrapping the Unit Circle
2
Essential Question What are the graphs of the
sine and cosine functions?
Enduring Understanding Know the characteristics
of the sine graph and cosine graph.
3

• Materials needed
• Paper
• Compass
• Protractor
• Ruler
• Markers
• Pencil

4
Fold your paper lengthwise. Then open.
5
Approximately 2-3 cm from the left edge of the
paper draw a tick mark on the fold.
6
Open your compass to a radius of 10-15 cm.
10 -15 cm
7
With your pencil point on the tick mark and the
compass point on the fold line, draw a circle.
Use your pencil to mark a dot in the center of
the circle.
8
Using your ruler draw both a horizontal axis and
a vertical axis intersecting at the center of the
circle. Extend both axes 1-2 cm beyond the edge
of the circle.
9
Since this is a unit circle, make a tick mark at
each intersection point of the axes and the
circle. Then label the tick marks accordingly.
1
1
-1
-1
10
Using your protractor, make a small tick mark
every 15 from 0 to 360 around the circle.
Label the tick marks on the outside of the circle.
1
45
30
15
1
-1
0
-1
11
Approximately 2 cm from the right side of the
horizontal axis, draw a 48 cm horizontal line on
the fold line.
1
1
-1
-1
On the left side of this line, draw a vertical
line at least the same length as the vertical
axis on the circle.
12
On the new vertical axis, draw a tick mark for 1
and -1 directly across from those on the vertical
axis of the circle. Label similarly.
1
1
-1
-1
-1
13
Beginning at the intersection point of the new
axes, draw a tick mark every 2 cm and label in
15-increments from 0 to 360.
1
1
1
-1
0 15 30 45 60 -------------------------------
-----------------------------------------------360

-1
-1
14
The graph of the sine function.
Remember that on a unit circle, the sine of the
angle is the vertical of the reference triangle
15
For each angle on the circle, using your ruler
measure the vertical length from the horizontal
axis to the point corresponding to that angle.
1
1
45
30
15
1
-1
0
0 15 30 45 60 -------------------------------
-----------------------------------------------360

-1
-1
Then draw a line segment the same length above
the corresponding tick mark in the coordinate
plane on the right.
16
Continue this process for each angle of the
circle from 0 to 360.
1
1
45
30
15
1
-1
0
0 15 30 45 60 -------------------------------
-----------------------------------------------360

-1
-1
Notice that after 180 the vertical segment is
below the horizontal axis indicating a negative
value. The segment should be drawn similarly in
the coordinate plane on the right after 180 as
well.
17
After you have completed the process from 0 to
360, use your pencil to smoothly connect the
dots at the far ends of the segments
1
1
1
-1
0 15 30 45 60 -------------------------------
-----------------------------------------------360

-1
-1
This curve is the graph of the sine function.
18
Sketch the Sine Graph below.
1
-360 -270 -180
-90 0 90
180 270 360
-2? -3?/2 -?
-?/2 0 ?/2
? 3?/2
2?
-1
Domain Range
19
The graph of the cosine function.
Remember that on a unit circle, the cosine of the
angle is the horizontal of the reference triangle
20
For each angle on the circle, using your ruler
measure the horizontal length from the vertical
axis to the point corresponding to that angle.
1
1
45
30
15
1
-1
0
0 15 30 45 60 -------------------------------
-----------------------------------------------360

-1
-1
Then draw a vertical line segment the same length
above the corresponding tick mark in the
coordinate plane on the right.
21
Continue this process for each angle of the
circle from 0 to 360.
1
1
45
30
15
1
-1
0
0 15 30 45 60 -------------------------------
-----------------------------------------------360

-1
-1
Notice that after 90 to 270 the segments are to
the left of the vertical axis. These segments
represent negative values so the segments should
be drawn below the x-axis in the coordinate
plane on the right.
22
After you have completed the process from 0 1o
360, use your pencil to smoothly connect the
dots at the far ends of the segments
1
1
1
-1
0 15 30 45 60 -------------------------------
-----------------------------------------------360

-1
-1
This curve is the graph of the cosine function.
23
Sketch the Cosine Graph below.
1
-360 -270 -180
-90 0 90
180 270 360
-2? -3?/2 -?
-?/2 ?/2
? 3?/2
2?
-1
Domain Range