Two Approaches to Trigonometry

Trigonometric Functions

Trigonometry is the study of the relationships between angles and sides of a triangle. This is the way it is introduced in secondary classes. The basic idea here is to use the concept of "similarity" of two triangles in a slightly formalized way and use it for practical applications like "heights and distances".

The theory begins by studying right-angled triangles and then generalizes to arbitrary triangles. The following figure captures the concept of "similarity" as used in trigonometry:

Fig 1. Similar Triangles

The idea here is to keep track of the ratio AB/BC and use it to calculate the length DE provided we know EF. Where trigonometry really gets smart of geometry is that it keeps track of the ratio AB/BC for all kinds of right-angled triangles. The shape of a right-angled triangle (not its size, we are talking about similar shapes) depends upon its angles, one of which is a right angle and other two sum upto another right angle. So specifying the smaller of the acute angles (in Fig 1, angles at vertices C and F for example) determines the shape of the triangle (and thus the ratio of various sides) completely.

Thus we define the trigonometric ratios (t-ratios in short) for an acute angle $ \theta$ by drawing a right-angled triangle one of whose acute angles is $ \theta$ as follows:

Fig 2. Trigonometric Ratios

The power of trigonometry lies in the fact that we can systematically calculate the t-ratios for many angles mathematically without doing any measurements. Secondary class textbooks provide the calculation for angles of values $ \theta = 30^{\circ}, 45^{\circ}, 60^{\circ}$ and its reasonably easy to handle these angles.

Then the addition formulas like (there are a large number of them) the following: $$\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$$ are used to get the values of the t-ratios for other angles. It is remarkable that with the help of these trigonometrical identities we can find the t-ratios for $ \sin 1^{\circ}$. However does anyone seriously believe that the tables of values of t-ratios for various angles are prepared using these identities? No!! That's really difficult given the amount of labor involved and its not systematic either. We will get back to the real story behind these calculations somewhat later.

To summarize, once we have the table of t-ratios, we can really apply the theory of t-ratios to solve engineering problems (the "heights and distances" problems are just tip of iceberg). The theory also involves lots and lots of identities (which are presented in intermediate classes) and techniques which we will not delve into.

However, if we look at these problems from a mathematical point of view, we are merely trying to calculate the length of some line segments and this does not seem like it should require so heavy mathematical machinery. All this heavy machinery of t-ratios is nothing more than the elementary geometry albeit done in a very systematic way (in fact, all the trigonometric identities can be derived using concept of similar triangles). So in reality, deep down the subject is more like a branch of geometry and there is nothing which is fundamentally new.

Or is there something really new in these t-ratios?

Circular Functions

We now move to solve a completely different but much more famous problem: the problem of "rectification" of a circle. The problem of measuring the circumference of a circle is a historical problem and it was known centuries ago that the circumference of a circle bears a constant ratio to the length of its diameter. The ratio is perhaps the most famous mathematical constant among laymen and is denoted by the Greek letter $ \pi$. In ancient times people tried to measure the circumference by approximating it with the perimeter of inscribed polygons and in the process calculated approximations to $ \pi$. Later on it was observed that it was actually very difficult to measure lengths of curves directly and people started focusing on lengths of line segments which were related to lengths of curves in a particular fashion.

We now illustrate this approach where we assume that any arc of a circle has a length which can be measured, at least theoretically:

Fig 3. Circular Functions

Here we are using language of coordinate geometry and we draw a circle of radius 1 with center at origin and let's say the circle cuts the positive X-axis at A (1, 0). Given any real number $ \theta$ we walk over the circle starting from point A and reach a point P such that the length of arc AP is $ \theta$. If $ \theta$ is positive then we walk in anti-clockwise direction, otherwise we take the clockwise direction. It is obvious that if we walk too much (say a distance of $ 2\pi$, then we reach back at A) we would loop around the circle many times. Therefore the final position of point P on the circle is periodic with period $ 2\pi$.

The co-ordinates of point P are defined to be $ \cos\theta$ and $ \sin\theta$. These we call the circular functions (as they are intimately connected with the arc length $ \theta$). The problem now gets simplified as measuring the co-ordinates (which are line segments) of P is simpler compared to measuring arc length AP (i.e $ \theta$, which is curved).

Using these definitions we can establish the addition formulas in particular (and thus all the other trigonometric identities). The circular functions are then studied rigorously using the methods of calculus and we finally get the following fundamental formulas: \begin{align} \cos \theta &= 1 - \frac{\theta^{2}}{2!} + \frac{\theta^{4}}{4!} - \frac{\theta^{6}}{6!} + \cdots\notag\\ \sin \theta &= \theta - \frac{\theta^{3}}{3!} + \frac{\theta^{5}}{5!} - \frac{\theta^{7}}{7!} + \cdots\notag\\ \theta &= \tan\theta - \frac{\tan^{3}\theta}{3} + \frac{\tan^{5}\theta}{5} - \frac{\tan^{7}\theta}{7} + \cdots\notag \end{align} the last formula being true only for the case $ -\pi / 4 \leq \theta \leq \pi / 4$.

The beauty of the above results lies in the fact that they give us a direct relation between length of curves ($ \theta$) and length of some line segments ($ \cos\theta$ for example) and this relation can be used to calculate one from the other in a very systematic way (nowadays programmed on a computer). It is indeed very strange that these curves can be rectified just by theoretical manipulation to yield such nice results.

It is this approach to trigonometry which actually makes it much more powerful than elementary geometry but the development in this area is considered more to be a part of calculus and we normally use the term "circular functions" instead of "trigonometric functions".

By now the reader must have got the idea as to how the tables of trigonometric ratios are obtained. The angles in degrees are converted into radians and then the series expansion above are used to compute the values of the trigonometric functions.

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