Basic Trigonometric Principles

Trigonometry, in the most basic sense, is the study of triangles, and specifically right triangles (those with one angle of exactly 90 degrees). While it may seem on the surface as if there could not possibly be very much to say about such simple objects (shouldn't basic knowledge of the Pythagorean Theorem be enough?), discovering the truth about these things is necessary in that it forms the basis of much more advanced mathematics, specifically in calculus.

Sines, Cosines and Tangents

These three terms form the most basic trigonometric principles. While finding the values for any of these for any given triangle is actually incredibly easy these days (thanks to the very convenient fact that almost every scientific calculator in the world possesses keys which do this very thing automatically), it can be easy to forget exactly what these words actually represent.

In essence, sines, cosines and tangents merely refer to the relationship, or ratio between the lengths of the various sides of any given right triangle, which varies depending only on the size of either angle (other than the right angle) in these triangles.

To understand where the values of these ratios come from, one must first begin by picking one of the angles of a triangle to use as a reference angle, which is thereafter usually denoted by the Greek symbol q (theta). Sine is then measured as the ratio of the side opposite angle q and the hypotenuse (which is the side opposite the right angle). Cosine is the ratio of the adjacent side and the hypotenuse, and tangent is the ratio of the opposite and the adjacent.

Now, providing one does not have a calculator handy, how can the values of these be discovered for any given angle, q? Unfortunately, the answer to this is probably not the one that most people are hoping for. Without a calculator (or the vast and detailed charts that were used to find these values for very specific angle sizes prior to electric calculators), the only way to find the sine, cosine or tangent values for an angle is to actually perform the measurements themselves. It was by doing so that the original table of values was originally developed, and this represented a great mathematical achievement which would prove invaluable to mathematicians (and scientists and any others who use these principles).

The Inverse Tools

In addition to these three ratios used to find the lengths of sides of right triangles, mathematicians also benefit greatly from knowledge of three other ratios: Secant, cosecant and cotangent.

These tools are used in a similar way to the first three, and likewise simply represent certain important ratios for use in right triangles.

The secant is found by first finding the sine of a given angle q. The secant can then be defined as 1/sine (an inverse of any number x is simply 1/x). Likewise,the cosecant is simply 1/cosine and the cotangent, of course, is 1/tangent.

Using Trigonometry

Where is trigonometry used? Obviously, it is not too often that the average person will find themselves in need of finding the ratio of sides in a right triangle (if they do, they surely have very interesting jobs), so why is any of this even necessary, apart from a purely mathematical curiosity?

The answer to this lies mainly in calculus. Trigonometric functions, though based on triangles, are actually used primarily in the study of patterns of points in a given circle (specifically, it is a circle called the unit circle, so called because it has a radius of exactly 1 unit). By knowing basic trigonometry, one has the tools necessary to understand how different points on the unit circle relate to one another, which actually leads (though the details won't be explained here), to an understanding of how certain wave functions can be calculated. This, in turn, leads directly to a great many physical applications, which is why trigonometry and calculus have become so invaluable to scientists over the past several centuries.

So while it may seem rather trite and pointless at the beginning, trigonometry is a vital tool in mathematics, and certainly not to be understated.

The Unit Circle

Print Article

Email Article

View All Articles

How to subscribe to feeds