## why trig is hard

Each year millions of students around the world struggle to learn trigonometry. They wrestle with complicated identities, strange functions related to circles, many special values, often involving that mysterious number \(\pi\). All that to measure triangles?? When they want to make a calculation, inevitably they need to resort to their calculator. Most of the students memorize the formulas, pass the tests, and then quickly forget all about the unpleasant experience. A minority enjoy the subject and perhaps go on to study mathematics or science at university. A much larger majority drop out or fail the subject.

And mathematicians wonder why so many people are turned off by mathematics??

There's a good reason that students find the current treatment so tedious and painful---it is the wrong way of thinking about the subject. Classical trigonometry represents a misunderstanding of the true nature of geometry. Up till a few years ago, there was however no alternative, so educators grimaced and plowed their way forward. Now however, there is rational trigonometry---in which the clouds roll away, everything becomes much, much simpler, and the natural beauty of the subject shines through.

It turns out that rational trigonometry has been there for the taking for two thousand years. All that was necessary was to go back to the ancient Greeks, and to take seriously the main creed of the Pythagorians: everything in the universe comes down to proportions between natural numbers.

And mathematicians wonder why so many people are turned off by mathematics??

There's a good reason that students find the current treatment so tedious and painful---it is the wrong way of thinking about the subject. Classical trigonometry represents a misunderstanding of the true nature of geometry. Up till a few years ago, there was however no alternative, so educators grimaced and plowed their way forward. Now however, there is rational trigonometry---in which the clouds roll away, everything becomes much, much simpler, and the natural beauty of the subject shines through.

It turns out that rational trigonometry has been there for the taking for two thousand years. All that was necessary was to go back to the ancient Greeks, and to take seriously the main creed of the Pythagorians: everything in the universe comes down to proportions between natural numbers.

## Calculus required for angles

The precise definition of an angle between two rays is as a ratio \[ \theta = \frac{l }{r} \].

where \(l\) is the length of the circular arc and \(r\) is the radius of the circle. As Archimedes realized, the definition of a circular arc requires ideas of calculus. He computed the arclength of an entire circle by approximating the circle from inside and outside by regular polygons and computing the perimeters of these polygons. In the limit as the number of sides of the approximating polygons increase, we approach the length of the circle.

This idea of taking the limit of an succession of increasingly accurate approximations is the heart of calculus. What Archimedes showed us is that calculating arclengths is really a problem of calculus.

There are other difficulties with angle besides the need for calculus. One concerns units. Should one measure in degrees, or in radians, or in grads as they do in some places in Europe? While the mathematics is simplest when we use radians, this system is not very intuitive, and results in horrible numbers for even the most common angles. An equilateral triangle has angles in radians of 1. 047197551 196597…. Also should we distinguish between angles measured clockwise and those measured anti-clockwise?

When it comes to lines, the reliance on circular arcs becomes confusing, since two lines divide a circular into four circular arcs. To work effectively with angles, we end up needing a lot of other transcendental functions from calculus, such as the cosine, sine and tan functions, as well as their inverse functions. This is a lot of machinery, just to study a simple triangle!

In a nutshell, the reason that the standard trigonometry is so complicated is because it unnecessarily confuses the study of triangles with the study of circles. These are separate mathematical subjects. There is no need for a technology that incorporates both together. I like to think of an analogy involving your backyard. Perhaps you have a lawnmower and a barbeque. Wouldn't it be great if we could have one machine that did both jobs at once??!! That way after you finished mowing your lawn you could just flip over the top and plop on your steaks! Well you can see what the problem would be; the thing would be unnecessarily cumbersome and expensive.

In just the same say, it is much more efficient and elegant to keep distinct subjects separate. For understanding triangles, no need of circles or circular functions are necessary. To understand circular motion, one doesn't need all the current song and dance, especially involving the reciprocal and inverse circular functions.

where \(l\) is the length of the circular arc and \(r\) is the radius of the circle. As Archimedes realized, the definition of a circular arc requires ideas of calculus. He computed the arclength of an entire circle by approximating the circle from inside and outside by regular polygons and computing the perimeters of these polygons. In the limit as the number of sides of the approximating polygons increase, we approach the length of the circle.

This idea of taking the limit of an succession of increasingly accurate approximations is the heart of calculus. What Archimedes showed us is that calculating arclengths is really a problem of calculus.

There are other difficulties with angle besides the need for calculus. One concerns units. Should one measure in degrees, or in radians, or in grads as they do in some places in Europe? While the mathematics is simplest when we use radians, this system is not very intuitive, and results in horrible numbers for even the most common angles. An equilateral triangle has angles in radians of 1. 047197551 196597…. Also should we distinguish between angles measured clockwise and those measured anti-clockwise?

When it comes to lines, the reliance on circular arcs becomes confusing, since two lines divide a circular into four circular arcs. To work effectively with angles, we end up needing a lot of other transcendental functions from calculus, such as the cosine, sine and tan functions, as well as their inverse functions. This is a lot of machinery, just to study a simple triangle!

In a nutshell, the reason that the standard trigonometry is so complicated is because it unnecessarily confuses the study of triangles with the study of circles. These are separate mathematical subjects. There is no need for a technology that incorporates both together. I like to think of an analogy involving your backyard. Perhaps you have a lawnmower and a barbeque. Wouldn't it be great if we could have one machine that did both jobs at once??!! That way after you finished mowing your lawn you could just flip over the top and plop on your steaks! Well you can see what the problem would be; the thing would be unnecessarily cumbersome and expensive.

In just the same say, it is much more efficient and elegant to keep distinct subjects separate. For understanding triangles, no need of circles or circular functions are necessary. To understand circular motion, one doesn't need all the current song and dance, especially involving the reciprocal and inverse circular functions.

## Ancient Greek geometry

The notion of spread can be motivated by going back to Thales, the teacher of Pythagoras and one of the key figures in the development of mathematics. Thales realized that the inclination of a line could be measured by the ratio of two quantities. He worked with distance.

The king of Egypt wished to know how high one of the pyramids was. He called upon the visiting Greek scholar Thales, (600 BC) who solved the problem with a simple but important argument, which has ramifications even to the classrooms of the twentieth century. Thales realized that if we consider the shadow of the pyramid when the sun is low on the horizon, then the proportions between the height of the pyramid to the horizontal length from the point directly below the apex to the shadow was equal to the corresponding proportion between the height of a stick in the ground and the horizontal distance from the stick to its shadow. In terms of an equation, we have h₁:d₁=h₂:d₂. Thus for example if d₁ is measured to be

Thales studied widely in the ancient world, and is attributed to a number of geometric results, including the fact that an isosceles triangle (with two equal sides) has also two equal angles, that a diameter of a circle bisects it, and that any angle subtended by a semicircle is a right angle. What are the meanings of these results?

An isosceles triangle is by definition a triangle with two (or more) sides equal. An angle in this context means the geometrical configuration consisting of two intersecting lines. When Thales stated that the angles at A and B were equal he meant that we could superimpose one upon the other, by a rigid motion of the plane. He did not mean that the measure of the angles were equal, a later concept that developed from the astronomers (who were very interested in measurements).

This proportionality that Thales discovered is an absolutely fundamental fact about the geometric world around us. Perhaps it seems too simple for such a grand claim? Far from it. We will see that its consequences are many.

Suppose l is a line making some inclination with the horizontal. Perhaps l is descibed as in the figure by going right 5 units and up 2. Then it is the proportion between these two numbers that determines the inclination of the line. The proportion 2:5 of rise: run is now called the slope of the line, and it is unchanged if we measure rise and run at a different point---this was the key observation of Thales. The Egyptians in their constructions of the pyramids used the reverse proportion of run: rise, which would be 5:2. However both of these concepts have a fatal flaw when applied systematically to geometry. When you view them not as proportions but as fractions, then there are situations where the denominator becomes zero. In the case of slope, a vertical line would yield a slope of 1:0 which gives the fraction 1/0 which is undefined, or `infinite'. In the case of the Egyptian proportion, the quantity associated to a horizontal line would be undefined.

The way out of the dilemma is to realize that the right triangle we are implicitly using has three quantities associated to it, and so far we are only using two. The third side is the hypotenuse, and it has the lovely property that it is never zero. So if we are going to create ratio, why not but the length of the hyptenuse in the denominator.

In our example, the length of the hypotenuse of a right triangle with perpendicular sides of lengths 5 and 2 is √(29). So we could possibly use either of the ratios

<K1.1/>

<K1.1 ilk="TABLE" >

(5/(√(29))) or (2/(√(29)))

</K1.1>

to describe the inclination of the line l. However we saw already that the Pythagorians were nervous about irrational lengths such as √(29) and this is why Euclid avoided distances in his work.

The solution to our dilemma you will now possibly have guessed: we should use ratios of quadrances instead of ratios of distances! So the two alternatives are

<K1.1/>

<K1.1 ilk="TABLE" >

s=((25)/(29)) or c=(4/(29)).

</K1.1>

We call s the spread between the two lines l and the horizontal, and c the cross between the two lines. For reasons that will become clearer as we go on, the spread will be treated as the more basic ratio, and we will state most of our theorems in terms of spreads. However there is because of Pythagoras theorem a simple relation between spread and cross that allows us to restate a law involving one to another:

s+c=1.

So the general definition of the spread between two lines l₁ and l₂ is: let A be the intersection of the two lines, let B be any point on l₁ and C the foot of the perpendicular from B to the second line l₂. Then

s(l₁,l₂)=((Q(B,C))/(Q(A,C))).

The spread is a ratio of two quadrances, so is a dimension-less quantity. There is no question about units! When the two lines coincide, then B and C coincide, so s(l₁,l₂)=0. We use this to define the spread between any two parallel lines to be 0. When the two lines are perpendicular, then C coincides with B, so s(l₁,l₂)=1. Any other situation is between these two extremes, so in general the spread between two lines is a number between 0 and 1.