In their effort to be mathematically precise, mathematicians use symbolism in ways that do not necessarily facilitate understanding. Take Einstein's famous formula E=mc^2. You may know what the letters mean and how to get a value for E, but do you really understand this formula and how its derived? In the same way, many students can "pretend to know" mathematics by applying formulas on homework and exams. These students miss an opportunity to genuinely understand the meaning behind the symbols and to experience the power of real problem solving. This article attempts to shed a little conceptual light on the teaching of a small segment of trigonometry - a topic that is most vulnerable to being used, but not understand.

You can always "pretend to know" mathematics by rotely plugging numbers into a formula and getting the right answer. But if you do, you miss an opportunity to understand what is going on. This article reveals the secret behind those mysterious trigonometric tables.

Learning without meaning
Here is a scenaro of a typical trigonometry lesson. A group of students are asked to measure the height of a pole (on the left) which is not accessible to direct measurement. They began by making a device known as a clinometer, which is really a protractor with a piece of weighted string attached to it. Their specific instructions for the activity were as follows:

Mark off a distance of 10 meters from the pole. This distance is called the baseline. Get an angle reading with the clinometer. Subtract this reading from 90. This is your angular distance. From a table of tangents find the number that corresponds to your angle. Multiply this number (the tangent of your angle) by the baseline. (Example: If your angle of elevation is 55 degrees, then the tangent is 1.42. The baseline is 10. Therefore the height of the pole is 1.42 x 10 or 14.2 meters.)

The students successfully plugged their values into the formula, got the right answer, then moved on to the next problem. No one bothered to ask why the product of the tangent of the "angular distance" and the baseline results in an estimate for the height of the pole! So what is this tangent? Does it have anything to do with what the students already know? The secret lies in knowing what this table is all about. It is time to add some meaning to the activity. On the right is a spreadsheet version of the table of tangents.

Pole measuring activity revisited
Here is another scenario.

In a science class a group of students are planning to build rockets that they will be launching later in the week. They want to be able to determine whose rocket will go the highest. The teacher suggested that they use a clinometer. They first learned how to use this instrument by measuring the height of a telephone pole across the street. The teacher demonstrated the use of the clinometer took the group outside and showed them what they would do. She measured off 10 meters from the pole and used the clinometer to measure the angle of elevation. Working in pairs (one measuring and the other reading) the students took two readings, recorded their results, and came back to class the next day with the results. The students agreed that 21 degrees was a reasonable clinometer reading. After some discussion they decided to make a scale drawing of the scene.

A few minutes later, one of the students realized that 21 degrees could not be the angle of elevation because it did not look right. Then they realized that the clinometer reading gave them the measure of angle ACB not CAB! But how do you find angle CAB? Noting that her students were stumped, the teacher directed their attention to a "dynamic" right triangle that was on the computer screen. What made the triangle dynamic was that the teacher could change the triangle by dragging a vertex to a different position which changed the measure of the acute angles.

A dynamic right triangle created in the Geometer's Sketchpad

She first established that this triangle was indeed a right triangle by noting that angle ABC was 90 degrees She challenged the students to see if they can "mess up" the triangle's "rightness" by moving the vertices to different positions. After some effort, the students realized they could change the lengths of the sides and the measures of angles BAC and ACB, but not angle ABC which remained 90 degrees. Next she made angle BCA equal 21 degrees. The students noticed that CAB became 69 degrees.

Finally, the triangle looked the way it was supposed to. The teacher focused their attention on how to figure out angle CAB if they knew the measurement for angle ACB which is the angle the clinometer measured. By dragging ACB and watching the measures of ACB and CAB change, they realized that the sum always remained 90.

Returning to their scale drawing they were able to come up with the unknown height. They solved the problem, but were not happy. They did not relish the thought of having to make scale drawings for every rocket that went up. There had to better way. Next, the teacher had them look at a digital image of the scene taken by a Quicktake 150 camera and copied as the PICT file into The Geometer's Sketchpad. The students discussed the relationship of this image with its "real" counterpart. They realized that the digital snapshot was really a scale drawing of the real triangle and could be measured just like their scale version. This time the teacher uses the Sketchpad's measuring tools.

Next the teacher wanted to know how much bigger the real triangle was than its Sketchpad counterpart. She discusses with them how the small triangle can be stretched to make the larger triangle. It turns out that whatever it takes to stretch 1.4 to equal 10 will also be needed to stretch the longer leg of the small triangle (3.6 inches) of the small triangle to become the unknown height.

The teacher explained that this "stretch factor" was the number of times that the real baseline was longer than its respective leg on the smaller triangle. Thus, the students divided 10 by 1.4 to get the ratio, but they all agreed that this process was really "ugly" way to arrive at an answer. Once again they wanted an easier way.

As it turns out, there is an easier way. And here is where the trigonometry comes in.

The tangent to the rescue

The problem would be easier if the enlargement factor was a whole number and if there was a quick way to the get the measurement of the side corresponding to the pole for any angle of elevation. Or, in other words, we need a "library" of measurements for every possible angle and an enlargement factor that is easy to determine and apply. The teacher told the students that a "library of tangents" would help them. The teacher began by defining a tangent: A tangent is a line that touches a circle in one and only one place. The students were then shown how to draw a tangent using The Geometer's Sketchpad

1. Draw a unit circle. (diameter is one unit.) 2. Draw a radius AB. 3. Highlight line AB and point B. 3. From the construct menu choose perpendicular line. 4. Place a point C on the line. 4. Hide the line. 5. Draw line segments AB and BC. 6. Measure angle BAC and segment BC.

It turns out that the measure of BC is the tangent of angle CAB. But how does this help us? First, this triangle is similar to the one we are trying to solve in our problem. Second, the radius BA equals 1, so the length of the baseline is the stretch factor. Third, we can find any angle measure we need by dragging vertex ACB up or down until it becomes the desired angle. Then the measure of BC becomes the length that is multiplied by the stretch factor, which gives you the length of the missing side.

Now the students could compare the measurements this triangle generates with the values in the tangent table (above.) The table of tangents can be thought of as a library of triangles with the measurement of the tangent line for the given angle.

Notice that our enlargement factor is now easy to determine since the side of the small triangle corresponds to the baseline and is always 1. This means that the baseline is always equal to the enlargement factor! After the students got their tangent length to correspond to the angle of elevation, all they needed to do was multiply the tangent times the baseline length, and they had their answer. And now, as Paul Harvey likes to say, you know the rest of the story.#