Mathematical Investigations III
Name
Mathematical Investigations III
Trigonometry- Modeling the Seas
HANDS-ON GRAPHING EXPERIENCE
THE SINE CURVE
Materials:
A circle with radius of 1 decimeter (a unit circle) and center marked by the intersection of
two perpendicular diameters (one black and the other red), two lengths (about 65 cm
each) of string or cord that will not stretch, 5 pieces of computer paper (not separated),
and a straight edge (for drawing axes).
Directions:
1.
Create The Graph.
Working with a partner, draw perpendicular axes centered in the middle of the five sheets
of computer paper. Label the longer axis that crosses all five sheets of paper the x-axis
and the shorter one the y-axis.
The right-hand endpoint of the black diameter of the circle should be labeled A. Using
the string, one member of the team measures an arbitrary length around the circumference
of the circle in a counterclockwise direction beginning at point A. The second member of
the team measures using the string from the point on the circumference where person #1
stopped straight down to the horizontal (black) diameter (i.e. measure the vertical
distance from the endpoint of the arc measurement to the axis from which the
measurement started). The first person might wish to start with arcs less than 90o.
string
(#1)
measure height with string
(#2)
A
DO NOT MAKE ANY MARKS ON THE STRING OR CIRCLES. THESE NEED
TO BE USED FOR MANY CLASSES.
Trig. 1.1
Rev. F10
Mathematical Investigations III
Name
With each partner holding their respective measured length on the strings, partner #1
marks off her measurement on the computer paper along the positive x-axis beginning at
the origin. Partner #2 marks off his measurement straight up from partner #1's mark.
Thus, the first point on the graph should have an x-coordinate equal to the length of
partner #1's measurement and a y-coordinate equal to the length of partner #2's
measurement. If the initial measurement was less than 90o, the point should fall in the
first quadrant.
put dot here!
string #2
string #1
NOTES: If the partner #1 measures in a counterclockwise direction from point A, then partner
#1 measures in a positive direction along the x-axis. If partner #1 measures in a
clockwise direction from point A, then partner #1 measures in a negative direction along
the x-axis. If the end of the arc falls above the black diameter, then partner #2 measures
in a positive direction parallel to the y-axis. If the end of the arc falls below the black
diameter then partner #2 measures in a negative direction parallel to the y-axis.
Repeat this process many times to generate more points of the graph in all quadrants.
Then sketch in the graph by joining points to make a nice smooth curve.
2.
Find Points On The Graph.
Consider how the curve was created and how each point was plotted. Use that
information to find and label the exact coordinates of as many points of the graph as you
can. You should be able to find the coordinates of more than 30 points. By the end of the
class period, you should have your graph sketched and a few points (in addition to the
maxima minima, and zeros) labeled.
3.
Analyze the Graph.
With your partner, write out a complete analysis of the graph you generated with the
strings and disk. Be sure to address the following:
Domain & Range
Intercepts
Max. points and min. points
Symmetries (both line and point)
Asymptotes, if any
Coordinates of other spec. points.
Trig. 1.2
Rev. F10
Mathematical Investigations III
Name
HANDS-ON GRAPHING EXPERIENCE
COSINE CURVE
Materials: A circle with radius of 1 decimeter (a unit circle) and center marked by the
intersection of two perpendicular diameters, two lengths of string or cord that will
not stretch, 5 pieces of computer paper (not separated), a straight edge.
Directions: Repeat the previous exploration with the following change.
Partner #2 measures the distance from the end of the arc horizontally across to the
vertical (red) diameter, instead of straight down to the horizontal (black) diameter.
For the purposes of direction, if the end of the arc lies to the right of the red
diameter, then partner #2's distance is considered positive. If the end of the arc lies
to the left of the red diameter, then partner #2's distance is considered negative.
string #1
string #2
A
put dot here!
string #2
string #1
As with the previous exploration, repeat the process many times to generate points
in all quadrants and a sufficient number of points to generate a smooth curve.
Considering how the curve was created, find and label the coordinates of as many
points of the graph as you can. Give exact coordinates.
Summary: Write a short paragraph describing the curve. How do these curves differ from the
previous curves we have discussed in this class (exponential, logarithmic,
polynomial)? What special characteristics do these curves have? What about
maximum points, minimum points, zeros? What types of symmetry exists?
Trig. 1.3
Rev. F10