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Archaeoastronomy

Curriculum Unit on Trigonometry

Background

Trigonometry as it is used and taught today has little relationship to the cultural context in which it originated. In order to undertake sea voyages to distant shores, fifteenth century navigators needed maps. Getting accurate maps and charts required better ways of calculating angles and distances. Trigonometry emerged -- together with advances in technology and the desire for evangelization and trade -- from the emerging capitalist spirit in small Mediterranean kingdoms in the 1400s.

The trigonometry developed at that time allowed anyone with tables of the sine, cosine, and tangent functions and an accurate way to measure angle and distance to calculate unreachable angles, distances, and areas. It enabled Columbus to track his position and movement well enough to sail the 3500 miles west from Cadiz, in Spain, to Hispaniola, in the Americas. All this was done by the "right triangle trig" now usually given little emphasis in mathematics textbooks. (This is the trig where the sine of an angle in a right triangle is defined as the ratio of the side opposite that angle to the hypotenuse of the right triangle. The cosine ratio for that angle is side adjacent to hypotenuse, and the tangent's ratio is side opposite to side adjacent.)

In the late 20th century, however, map making depends upon aerial photography rather than trigonometry. And modern applications of trigonometry have moved on to modeling periodic motion, angular velocity and momentum, and solids of rotation.

In order to demonstrate the power of trigonometry to high school students, most teachers still use examples of historical, or "cultural" trigonometry. The class goes outdoors to calculate the height of a flagpole or a distance across an imaginary river. Or a surveyor tells the class how trig is used (or could be used) in triangulation. Meanwhile the meat of the course continues with graphing trig functions, solving trig equations, verifying identities, and analyzing relationship between the six trigonometric functions. The visual anchor no longer is the simple right triangle with its hypotenuse, opposite, and adjacent sides. In most high school math texts, the preferred visual model is a circle with its center at the origin of a coordinate (x-y) plane and with each point on the circle defined by an x- and a y-coordinate and r, the radius of the circle.

It appears that the archaeoastronomy of the U.S. Southwest implies connections with modern trigonometry that European historical trigonometry misses. Rather than angles and distances on earth for their own sake, the ancient Pueblo people concerned themselves with the movement of heavenly bodies across the sky, especially of the sun and the moon and especially with their rising in the east and setting in the west. In the living pueblos today, a religious leader has the important job of tracking sunrise or sunset points along the horizon in order to predict winter and summer solstices. There is ample evidence at sites in and around Chaco Canyon that their Anasazi predecessors did too. (Zeilik, 1997 and Malville, 1993)

This unit focuses upon the recording of horizon line sunrises (or sunsets) and the relation of that periodic motion to the cosine function. While doing that, the unit also invites students to examine the purpose and methodology of angular measurement as well as historical and ethnographic implications of geocentric and heliocentric conceptions of space.

A person who observes exactly where the sun rises notices that during most of the winter and spring the rising point moves northward every day, and during most of the summer and fall it moves southward. At the time of the solstices, however, the rising point of the sun doesn't appear to move at all for several days. In fact, the rate of change of the sunrise points varies throughout its cycle, moving fastest near the equinoxes and slowest (or not at all) near the solstices. Winter solstice is crucial to hunters and gatherers and agriculturalists living in mid-northern latitudes because it means that the sun ceases rising and setting further and further south (causing shorter and shorter days) and will begin moving northward again. Pueblo societies wanted to be able to predict this "turning around" of the sun, and they assigned this task to a sun priest.

For a sun priest in pueblo societies today (and presumably for the Anasazi of the 1200s), the goal is not only to determine when the solstices occur but also to be able to predict them far enough in advance so that preparations can be made for the ceremonies. There is no indication that Pueblo astronomers used any day counts greater than around fifty days. In other words, they don't predict the winter solstice by counting 365 days from the last winter solstice. What they do is to establish, over a period of a few years, a horizon mark for a date some time prior to the solstice. Then they are able to announce to their village, say eight days beforehand, when the solstice will occur, and preparations can begin for its observance. The entire procedure would be something like trying to locate the exact maximum of a sine wave, which can only be done with reference to the lesser values symmetrically located on either side of the maximum point. (See Suina, 1992, for cautions about interpreting Pueblo religious observances.)

Another method sometimes employed by Pueblo astronomers, especially when no adequate horizon line exists, is to use an opening in a building that casts a beam of sunrise against a wall opposite. Successive sunrises then appear to move (in reverse direction) across the wall. Turning points -- solstices -- can be marked with turquoise, shell, or bone, as can anticipatory dates a certain number of days prior to each solstice. The advantage of this kind of sun reckoning to archaeologists is that they may survive in the ruins of ancient buildings, whereas horizon marking leaves only occasionally some trace of the station from which the horizon was viewed. The caution regarding the verification of a prehistoric sunlight aperture for wall casting is that it may appear by chance in a reconstructed ruin when it may never have had any astronomical meaning to the original inhabitants.

For many people now in the teaching profession, the idea that Columbus "discovered" the New World is linked to the view that on his first voyage he set out (with sailors fearful of falling off the edge) to disprove the notion of a flat earth. And that when he reached the "Indies," even though he'd miscalculated his position by some 8,000 miles, he had forever demolished the Medieval theory of the flat earth and its geocentric underpinnings.

We expect now that students learn something about the fallacies inherent in the above paragraph. That Columbus happened upon a "new world," previously unknown to Europeans, that was rapidly moving from tribal to national social and political systems and which soon would be ravaged by diseases imported from Eurasia. That Columbus set sail, sponsored by Italian merchants eager to maintain commercial ties to China and India, in a time when the notion of a flat earth for practical purposes had long since been abandoned. And that the reason he has been portrayed as a bold, scientific discoverer, rather than as an oppressor and plunderer, has very much to do with the desire of Americans to feature the peaceful rather than the conquistorial nature of the European penetration of the Americas. (See Bigelow, Krupp, and Sanchez for extended discussion on the persistence of colonial perspectives on native thought and practice.)

It is also widely understood that the work of Copernicus (1473-1543) and Kepler (1571-1630) made possible the more sophisticated Western conception of the (heliocentric) solar system and its place among galaxies of distant stars. What may not be as widely understood is that the geocentric universe (sun, moon, planets, and stars revolving around the earth), supposedly consigned by Columbus to reliquaries of the Middle Ages, is actually the one modern astronomers use much of the time to track celestial bodies and events. Rising and setting, azimuth, zenith, angular size and displacement, scientific phrases all, still descend from a geocentric model as well suited for a modern astronomer as for the ancient Anasazi skywatcher.

The modern school year, just as the ancient Pueblo year, centers on the winter solstice. For a high school trig class this means that by the time the winter solstice arrives, there has been ample time to establish a horizon line and begin recording sunrise or sunset points along that line. It also means that if recordings are begun soon after school starts in September, students can learn and refine recording procedures during a time when daily intervals are at their maximum lengths. In many pre-calculus courses, little actual trigonometry is done during the fall semester; but horizon line recordings would be done then, and later analyzed during the more specifically trig sections of the course during the spring semester. (See "Beginning Activities" numbers 3 and 4 on pp 12-13.)

Even though the sine function may be more commonly understood, for consistency with students learning trigonometry the cosine works better. The cosine wave is identical to the sine wave except for being shifted 90 degrees to the left. But the cosine depends upon x values (horizontal values), and better models movement along a horizon line. In the activities that follow, the horizon line is treated as the x-axis. Angle values of zero and 180 degrees represent the winter and summer solstices, respectively, while 90 and 270 degrees represent the vernal and autumnal equinoxes. (See Figure 1.)

Wall castings follow a similar model, with the directions reversed. (See Figure 2 and "Beginning Activity" #5.) Basic instructions for classroom applications include: 1) cutting a slit (about a quarter inch wide and one inch high) in a piece of cardboard; 2) taping the cardboard to a window that faces the eastern horizon; and 3) setting a piece of masonite (or another cardboard) parallel to the window and a convenient distance inside the room. At a distance of four feet, the masonite will need to be at least 5 feet long to accomodate the extremes of the winter and summer solstices. At sunrise at the equinoxes, the slit in the window cardboard will project a rectangle of light directly opposite on the masonite. As the sun moves toward a solstice, the sunrise light moves north or south along the masonite until it approaches a standstill. The light projection loses its rectangular shape but still travels a measureable distance each day, except within a few days of the solstice. (Five feet divided by the 180-day half-year works out to an average daily change of one-third of an inch.) Just as for horizon marks, the record of daily changes on the masonite can be modeled by the cosine function, with fast movement at the equinoxes (90 and 270 degrees) and barely perceptible change at the solstices (zero degrees for the winter solstice, 180 for the summer).

Given the basic mathematical correspondence -- the cosine function that reduces circular motion to the x-axis, the sun's movement around the circle of the ecliptic reduced to movement along a horizon line -- there remain crucial practical considerations. Establishing a horizon line can be tricky. A preference between the views to the east and west may determine whether sunrises or sunsets will be recorded. A featureless horizon will serve no purpose for horizon line recording. Neither will a too distant, or a too near horizon. Because of perspective, distant sizes seem smaller than near ones. Distant movement is less perceptible than near movement; and features (peaks, notches, etc.) which would distinguish a nearer horizon become less distinct with distance. In order to be able to record the change in each morning's sunrise location, an observer must be able to perceive some change in location, but not so much that slight movements of the head or body may affect the results. The ideal of a horizon line that has a resolving power of from 4 to 8 minutes of arc three weeks before the solstice implies an ideal horizon of from 2 to 10 miles distant.

For summer solstice calculations, it happens that the Four Corners area, a region of heavy settlement by ancient Pueblo people where the modern states of New Mexico, Colorado, Utah, and Arizona meet, normally experiences exceptionally dry and clear weather preceeding the summer solstice and lasting for a week or two after it. (The summer rains often begin in early July.) This means that in doing the observations for a new pueblo in a new site, a sun priest might have about three weeks to verify that an early June rising point matched an early July reading, to establish the summer solstice exactly in between. In future years, the observer would know that the solstice would occur exactly half the total number of days between the two identical readings after the early June reading.

For an interior light casting, however, a flat horizon becomes ideal, for this reason: especially near the summer solstice, the sun rises at a sharply slanting angle; and this slant can be amplified by distant peaks and valleys to distort the true direction of the sun's rising point. Also for wall castings, the observer must strike a balance. The greater the distance between the slit that admits the sunlight and the recording wall, the more day-to-day movement there is to observe, but also the fuzzier the image becomes. An average of one cm of change per day seems ideal, and this implies an ideal distance of about one meter between the spot where light enters the room and the wall upon which daily changes are recorded.

Another difficulty in recording observations, of course, stems from the fact that sunrises and sunsets normally occur outside the typical school day. So the students are doing a new and strange activity at an often unusual time of day. However, students can practice sketching a horizon line at any time of day and from any location (someplace on the school grounds, perhaps) which affords a clear view of a horizon. In order to keep the drawing to scale, students can learn a basic sighting method which depends upon another application of "modern" trigonometry, that of radian measure. The outstretched fist has long been used by skywatchers to estimate the size of objects in the sky, angular distances between stars, and degrees of arc along the horizon. The basic calculation involves dividing the length of an arc by the length of its radius. This is the same relationship defined by angle measure in radians in the formula

0 = s/r

(where 0 is the angle, s the arc length, and r the radius).

The procedure is personal: (See Mini-project #7.)

1) Divide the width of your outstretched fist by the distance from your eye to your fist.

[This gives the angle, in radians, across your fist.]

2) Then multiply that result by 57.

[That converts radian measure to degree measure, since there are approximately 57 degrees to one radian.]

Once the degrees of the fist have been calculated, say 8 degrees, then students can impose that scale upon the horizon. Beginning at due east, they can divide the horizon into six to eight 8-degree segments. This results in a sweep of the horizon from about 30 degrees north to about 30 degrees south of due east.

Beginning Activities: Mapping, observing, and recording

Objectives:

1) students calculate the angular distance of their own outstretched fist

2) students apply radian measure to the calculation of angular distance

3) students locate and sketch a horizon profile appropriate for tracking sunrise (or sunset) points

4) students gain experience recording sunrise (or sunset) points along their horizon profile

5) students experiment with the trigonometry of light casting, and record daily changes to estimate solstices and equinoxes.

Carrying out the Beginning Activities

1. Students practice approximating angular measurements using their outstretched fist. They measure the distance s, across their knuckles and the distance r, from their eye to their fist. Dividing s by r gives the angle 0 in radians. Multiplying 0 by 57 gives an approximation of the degrees swept in the sky by their outstretched fist.

2. An alternative procedure uses the astronomers' formula

s = .018Ad

for approximating distances and diameters in space. Here s stands for a diameter or the distance between two celestial bodies, d stands for their distance from the observer, A the angle in degrees between the bodies, and .018 represents the conversion factor 1/57 that relates radians to degrees.

3. Students select a horizon line near their home and sketch it to scale using angular measurements of their outstretched fist.

Several considerations must be taken into account. Does the horizon have enough relief to show short distances? Is it visible winter and summer (with leaves on the trees)? Does it extend far enough north and south for the particular latitude? Even though it probably would not be used for sunrises or sunsets because of the time of day students are at school, a class project constructing a horizon line near the school is recommended so that students become familiar with procedures they may never have used before.

4. Students begin recording sunrise or sunset points and dates.

5. Students in the classroom experiment casting a small beam of sunlight upon a surface. In a morning class, students can simulate an actual sunrise by establishing a false horizon in a nearby roof or by taping a piece of cardboard partway up an east facing window. They then place two pieces of cardboard inside the classroom and parallel to the window. The first, nearest the window, has a small opening for projecting the sunlight. The second piece of cardboard is used for recording "sunrises" every day. Students begin experimenting with various distances between the pieces of cardboard to see which "length of throw" of the light produces the best results in tracking daily changes in sunrise points.

An optimal length of throw is one that reveals day to day change in sunrise locations with a sharp and easy-to-trace image. Once optimal procedures are established, students continue to record sunrise points throughout the semester.

Mini-Projects

Mini-projects are intended to give students the chance to figure out a problem and to explain how they did it. At the teacher's discretion, they are intended for teamwork or for students working individually. They may be written or presented orally (or some combination of the two). The basic components of any mini-project are:

1) Tell briefly and succinctly what the project is.

2) Show the steps you used to solve the problem. Include figures, diagrams, charts, and graphs as appropriate.

3) Explain what the answer means. Explain what you did to get it.

Mini-projects are assessed according to how well the above three basic components are done AND upon the accuracy of the answer. Mini-projects can be done together as part of the regular class assignment OR they can procede independently of other classwork. If they are presented orally to the class, then presentation skills (clarity, eye contact, quality of visuals, coordination among presenters) can be assessed additionally by students in the audience.

1. Circular Movement [See Figure 1.]

Objective: student correlates movement around a circle to movement across its diameter.

Materials needed: compass and protractor, horizon profile marked with sunrise points.

Procedures: Draw a large circle with a horizontal diameter. Beginning at the right end of the diameter, mark 15 degree intervals on the circle all around the circle. Drop perpendiculars from each point on the circle to the diameter. Describe how the perpendiculars move along the diameter. Compare that movement to the movement of sunset marks along a horizon line.

2. Cosine Values on a Number Line [See Figure 3.]

Objectives: students visualize change in cosine values

Materials Needed: ruler, calculator for cosine values

Procedures: Draw a long number line, labeled by tenths, with zero at the center, -1 at the left end and 1 at the right end. Calculate the cosines of all the angles at 15 degree intervals from 0 to 360 degrees. Plot those cosine values on the number line. Explain the connection to the sunset marks on a horizon line. Why is the cosine function used? What angle interval would be used to model the sunset marks every day?

3. Using TI-83 LIST options to Model sunrise points

Objective: Students see how the cosine function models a reflection of circular motion.

Materials Needed: TI-83 graphing calculator

Procedures: Use a TI-83 graphing calculator to mimic the movement of the sunset marks and to calculate the intervals between successive marks.

Set up Lists 1, 2, and 3.

STAT 5 then type L ,L ,L

Store 5-degree angle intervals to List 1.

seq(5X,X,1,72,1) L [seq is in 2nd LIST 5, is STO ]

Store the cosines of the values in List 1 to List 2.

[STAT 1:Edit place cursor on L and type cos (2nd L )]

How are the standstills modeled in List 2? What about the equinoxes?

Calculate the intervals between points in List 2 and store them as List 3.

[place cursor on L and type 2nd LIST OPS 7 2nd L ]

Questions: What do the intervals in List 3 have to do with the day-to-day change in sunset marks throughout the year?

What do they have to do with the slope at various points along the cosine curve? With maxima and minima?

Mini-projects 4, 5, and 6 extend concepts from #1, #2, and #3

4. What would happen if the daily sunset horizon marks were plotted on a number line as in mini-project #2 and the inverse cosine values were calculated? What would you expect as the interval between those inverse cosine values? Why?

5. Mini-project #1 models sunset marks on a horizon line without any reference to trigonometry, but only to a circle. How do you think that could relate to ancient Pueblo thinking, especially as compared to miniprojects #2 and #3?

6. If a student wouldn't tell you her birthday, but located it on a horizon line, to what accuracy could you predict her birthday?

Mini-projects 7, 8, and 9: Using your Fist and fingers

Objective: student learns to use his/her hand and fingers to measure celestial distances

Materials Needed: tape measure or ruler

7. Calculate the diameter-to-distance ratio of your outstretched fist. (See p. 9) Of your little finger. Of your thumb. Convert each ratio to an angular diameter using the formula

s/d = A/57 (or s/r = 0).

8. Use the angular diameter of your little finger to estimate the angular diameter of the daytime moon. If you know that the moon is about 2,000 miles in diameter, can you estimate its distance from the earth? (See also Zeilik, 1998, Interactive Lesson Guide for Astronomy. The first four "Focused Discussions, pp 11-18, all deal with angular sizes and distances.)

9. (During Balloon Festival) Compare the angular diameter of a balloon to the moon's. Estimate the distance to the balloon using 20 meters as an average diameter for a balloon.

[Use the formula r = s/0]

Mini-projects 10, 11, and 12: Thought Questions on the connections between Trigonometry and Astronomy.

10. Calculate the error involved in using angular diameter to approximate an actual diameter. On a graphing calculator:

y = x/180 gives angular diameter

y = sin x / sin .5(180-x) (from the Law of Sines) gives the linear diameter

Use the TABLE function to compare the two diameters for small values of x. Calculate the percent error at 5 degrees.

11. Why do you think astronomers, who do cutting edge scientific research, use degrees instead of radians?

12. Is the horizon a line or an arc? What difference does it make in calculating sunrise points?

         
Figure 1                                  Figure 2                                            Figure 3

Bibliography

                    Zeilik, Michael: Astronomy: The Evolving Universe, 1997