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The
study of trigonometry, which translates verbatim as "triangle measurement",
began more than 2000 years ago, partially as a means to solving land surveying
problems. The trigonometric functions we use today are not the same as those
used 2000 years ago. Early trigonometry related the length of a chord of
a circle as the value of a trigonometric function.
In a circle of a fixed radius,
the problem was to find the length of the chord subtended by a given angle.
For a unit circle, the length of the chord subtended by the angle x
is, y =  .
Although the first known tables
of chords have not survived, it is claimed that twelve books containing
these tables were written by the Greek mathematician Hipparchus, around
140 B.C. Due to this, it is claimed that he is the founder of trigonometry.
In the sixteenth century,
right triangles were used to define the trigonometric functions that we
are more familiar with today. We will use a modified right triangle approach
to define the trigonometric functions by placing one of the acute angles
of a right triangle on a coordinate plane. We will explore its measurements
using The Geometer’s Sketchpad.
Part 1:
- Open a new sketch and
define a coordinate system in your sketch window.
To define a coordinate
system:
- Select Define Coordinate System, under the Graph menu.
- You may drag the point (1,0) along the x-axis to define the size
of the scaling unit.
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- Construct
a unit circle centered at the origin; that is, circle with a radius
of one inch centered at the origin. Be sure to first change your Distance
Unit to inches in your Preferences window.
- Label or relabel the
point (1,0) as point D.
To
relabel an already labeled figure:
- Select the Text Tool
 from
the toolbox.
- Position the hand pointer
on any label.
- Double click and type the
new label in the Label dialog box.
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We would like to investigate
a group of right triangles, where (1) the length of the hypotenuse is
equal to the radius of the circle; (2) the vertex of the right angle is
confined to the x-axis; and (3) a vertex is at the origin.
- Using the three parameters
above, sketch a representative right triangle on paper. Describe and
compare your triangle with your neighbor's.
- Construct an arbitrary
point on the unit circle and drag this point around the circle until it lies
in the first quadrant. Re-label it as point B. Construct the radius
of the circle, by constructing a segment from the origin to point B.
This construction satisfies the first parameter, the length of the
hypotenuse is equal to the radius of the circle.
- Construct a perpendicular
line to the x-axis through point B. Construct the intersection point
of the line and the x-axis. Re-label it as point C. Construct a segment
between points B and C. Change the display of the segment BC to thick
and red. Hide the perpendicular line. This construction satisfies
the second parameter, the vertex of the right angle is confined to
the x-axis.
- Construct a segment from
the origin to point C. Change the segment's color to green. This
construction satisfies the third parameter, a vertex is at the origin.
- Observe the constructed
triangle on the inside of the circle. Label the sides of the
triangle in the conventional manner. Compare your sketch with the sketch
below. (Instructor
Note: A
Sketchpad
file illustrating the unit
circle has been saved as unitcircle.gsp.)
- What kind of triangle have
you constructed? How can you be sure?
- As you drag point B around
the circle, focus your attention on the lengths of sides a and b. Qualitatively
describe the lengths of a and b as point B is dragged around the circle.
Write down your observations.
- It is difficult to describe
specifics about lengths a and b without being able to pinpoint "locations"
around the circle. We shall use the measurement of the angle DAB to
refer to the location of point B on circle. Change your Angle
Unit to directed degrees in your Preferences window.
- Measure angle DAB. Observe
the angle measurement as point B is dragged around the circle. Describe
the angle’s measurement. Can you have an angle that has the measure
of 240° ? Why or why not? What does this tell you about The Geometer’s
Sketchpad measurement techniques? Reconcile the differences between
the angle measurement techniques of The Geometer’s Sketchpad and the
conventional unit circle.
- Measure the lengths of
sides a and b. At what point on the circle are lengths a and b congruent?
Drag point B around the circle to find where lengths a and b reach their
maximum and minimum lengths.
Part 2:
In your sketch, you notice
the lengths of the sides of the triangle changing as point B is dragged
around the circle. We would like to graph the length of side a as point
B is dragged around the circle and observe the patterns displayed. To
graph, we will be using both the animate and trace features
of The Geometer’s Sketchpad.
- Once again, drag point
B around the circle. On a sheet of paper, sketch a graph of the length
of side a as a function of the measure of < DAB.
- Using The Geometer’s
Sketchpad,
we can graph the length of side a versus the measure of angle DAB. We
would like to trace the length of side a as the measure of angle DAB
changes. Estimate the distance, in inches, point B will travel around
the unit circle once. Use The Geometer’s Sketchpad to measure this distance.
To measure
a circle’s circumference:
- Select the circle.
- Select Circumference from the Measure menu.
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- Mentally compare your estimate
to the computed circumference of the circle. Was your estimate reasonable?
Why or why not? To serve as the x-axis of the graph, construct a segment
the same length as the circle’s circumference starting at the origin,
along the positive x-axis. Construct a point on this new segment, and color
it blue. Drag the point along the segment to be sure it is confined
to the segment.
- This segment defines the
x-coordinates of our graph. In order to define the y-coordinates, we
would like to construct a point whose distance from the x-axis is the
same as the length of side a. Construct a dashed line parallel to the
x-axis through the point B. Construct another dashed line, parallel
to the y-axis through the blue point. Construct the point of intersection
of the two dashed lines. Change its color to green. Drag the
green point. Does its distance from the x-axis equal the length of segment
a?
- Describe what happens to
the green point as point B is dragged around the circle. Describe what
happens to the green point as the blue point is dragged along its segment.
We would like to trace the
green point while point B travels around the circle AND while the blue
point travels along its segment. However we cannot drag both at the same
time with our mouse....but we can have Sketchpad animate both points together.
- Create an animation button
to trace the green point as point B travels counter-clockwise around
the circle once slowly and while the blue point travels forward along
its segment once slowly.
To identify
which point to trace:
- Select the point.
- Choose Trace Point under the Display menu.
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To create
an animation button for two objects:
- Select the objects you wish to animate.
- Choose Action Button under the Edit menu, and
then Animation.
- In the Properties of Action Button window, make travel selections for
your object (i.e., its direction and speed).
- Click on the Label tab to rename your action button,
and click OK.
- An Animate button will appear in your sketch window.
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- Before you activate your
button, predict the shape of the graph by a brief sketch on your paper.
Activate your button by clicking on the Animate
button. To cease the animation, re-click on the Animate button.
- Describe the path that
the green point traced. Compare your predicted graph with that one in
your Sketchpad window. In what ways is it accurate? In what ways is
it inaccurate? What is the length of segment a when the measure of angle
DAB is 0°?
- We need to make sure we
have an appropriate starting point to trace the length of side a. What
would be an appropriate starting point? To start our tracing properly
we need to set the measure of angle DAB to 0° . How could we change
the angle measurement of angle DAB?
- Which points need to be
moved in order for your graph to be accurate? Move those points and
re-activate the animation button. How did your graph change? Reconcile
any differences between your predicted graph and the graph in your sketch
window.
To erase the
traces:
- Choose Erase Traces under the Display menu.
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- One way is to have
Sketchpad
move a point to another point’s location in your sketch window. This
can be done with a Movement button. Create a movement button
that will move point B to point D and the blue point to the origin.
To create
a movement button:
- Select the first point you would like to move; then select the
point where you would like it to move to.
- Select the second point you would like to move; and the point
you would like it to move to.
- Choose Action Button under the Edit menu, and then
Movement.
- Select the movement speed you desire and click OK.
- A Movement button will appear in your sketch window.
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- Click on the Movement
button in your sketch window. Describe what happens. Let’s give this
button a more descriptive name than "Move."
To re-name
a button:
- Using the Text Tool
double click on the button.
- Type in the window a more descriptive name.
- Then click OK.
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- Click on the animate
button in your sketch window. How has the graph changed? At what angle
measure(s) does the graph cross the x-axis? (Instructor
Note: A
Sketchpad
file illustrating the unit
circle and the sine function has been saved as sine.gsp.)
- If you change the radius
of your circle, predict how the graph will change. Change the radius
of the circle, reset your graph and double click on the animate button
and assess your predictions.
The function you have graphed
is called the sine function. On a historical note,
the Hindu word jya
for the sine was adopted by the Arabs who called the sine jiba,
a meaningless word with the same sound as jya. Now jiba
became jaib in later Arab writings and this word does have a
meaning, namely a ‘fold’. When European authors translated the Arabic
mathematical works into Latin they translated jaib into the word
sinus meaning fold in Latin. In particular Fibonacci's use of
the term sinus rectus arcus soon encouraged the universal use
of sine (http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Trigonometric
_functions.html, retrieved April 15, 1999).
Formally, the definitions
of cos θ and sin θ can be generalized as follows. If θ is an angle in standard position and if (x,y) is any point (other than
the origin) on the terminal side of θ, then
and  where 
- Using the calculator and
the lengths of the sides of the triangle, compute the sine and cosine
ratios. Compare those values with the calculator Functions sine/cosine
for the angle DAB.
- Drag point B around the
circle. How do the ratios and the function values compare? Are they
always equivalent?
Part 3:
In your sketch, we graphed
the length of side a as the measurement of angle DAB increases. We would
now like to do a similar investigation by graphing the length of side
b as the angle DAB changes.
- Drag point B around the
circle. On a sheet of paper, sketch a graph of the length of side b
as the measurement of angle DAB increases.
- Using a slightly modified
process as above, create another Animate button in your sketch
window that will trace the length of side b as the measurement of angle
DAB increases. You may want to change the color of the point being traced
to distinguish it from the sine function.
To trace
the length of side b as the measure of <DAB changes:
- Drag point B to the first quadrant.
- Drag the dashed vertical line to the right of the origin.
- Similar to side a, we would like to have a vertical segment the
same length as side b. To do this, rotate side b 90°
about the origin . Select the origin and Mark Center from the Transform
menu.
- Select both side b and its endpoints. Rotate the segment 90°
by choosing Rotate from the Transform menu.
- Construct a dashed perpendicular line to the rotated segment through
its endpoint.
- Construct the point of intersection between this dashed line and
the vertical dashed line. Change its color to red. Hide both the
rotated segment and its endpoint.
- Select the red point and Trace Point under the Display
menu.
- Now we are ready to animate. Select point B and
blue point along the x-axis.
- Choose Action Button under the Edit menu, and then
Animation.
- In the Properties of Action Button window, make travel selections for
your objects as follows: point B travels
counter-clockwise around
the circle once slowly and while the blue point travels forward along
its segment once slowly. Click OK.
- An Animate button will appear in your sketch window. Rename
the button to distinguish it from the other Animate button in your
sketch window.
- Click on the movement button to reset the points; and then
click on your new Animate button.
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- Describe your observations
when the button was activated.
- Qualitatively and quantitatively
compare and contrast the sine and cosine functions.
- For what angle measurements
are the sine function positive? Negative? For what angle measurements
are the cosine function positive? Negative? What are the roots of each
of the functions? How could you find the roots of the functions mathematically?
- For what angle measurement
does the sine function equal the cosine function?
Part 4:
- Both the red and green
points are traced when you click on either of your Animate buttons
resulting in both functions being graphed simultaneously. What
was the difference between the Animate buttons? How could you have only
one of the functions to be graphed?
- The Geometer’s Sketchpad
has a type of Animation button which allows you to show and hide
objects in your sketch window. Create Hide/Show buttons to hide
the traces for each of the trigonometric functions. (Instructor
Note: A
Sketchpad
file illustrating both
the sine and cosine functions has been saved as
trig.gsp.)
To create
a Hide/Show button:
- Select the point you wish to hide.
- Choose Action Button under the Edit menu, and then
Hide/Show.
- A button will appear in your sketch window. Rename the
button with a more descriptive name.
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- Click on the Hide
button you have created for the sine function. Click on the Animate
button. What changed in your sketch window? Experiment with the other
Hide/Show button.
- What are the advantages
and disadvantages to the Hide/Show buttons?
Extensions:
- Why was the unit
circle chosen to construct the sine and cosine curves?
- Using your unit circle,
construct the graphs of other trigonometric functions.
- Conjecture what the graph
would look like if point B traveled clockwise.
- Sketch a graph, by hand,
of the area of the triangle verses the angle measurement.
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