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 create axes in your sketch window.

• 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.

• Relabel point B as D in your sketch window.

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 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.

• 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.

• 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. Select a dark color for this segment to distinguish it from the 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 slowly once around the circle and while the blue point travels slowly one-way along its segment.

• Before you activate your button, predict the shape of the graph by a brief sketch on your paper. Activate your button by double clicking on the Animate button. To cease the animation, click anywhere in your sketch window.

• 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.

• 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.

• Double click on the Movement button in your sketch window. Describe what happens. Let’s give this button a more descriptive name than "Move."

• Double 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?

• 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 q and sin q can be generalized as follows. If q is an angle in standard position and if (x,y) is any point (other than the origin) on the terminal side of q , 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.

• 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 rotts of the functions mathematically?

• For what angle measurement does the sine function equal the cosine function?

• 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.

• Click on the Hide button you have created for the sine function. Double click on the Animate button. What changed in your sketch window? Experiment with the other Hide/Show buttons.

• 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.