Trigonometry is hailed as a rough subject, particularly for precocious high school students brave enough to wade into its potentially turbulent waters. Notwithstanding "trig" challenges, the key to mastering this subject really lies in the basic understanding of its two key players: the sine and cosine. All the other features of this fascinating subject really derive from the shenanigans that these two play on unsuspecting students. Consequently, before students plunge into this mysterious realm of triangular measurement and other identities, they should make sure they are intimately familiar with its two key players and the critical roles they play.

Trigonometry derives from the Greek "trigonon" = triangle and "metron" = to measure. In essence, trigonometry means the measure of triangles, and as such all the trigonometric functions derive from the triangle, specifically the right triangle; however, the true science encompasses a lot more than the simple measurement of triangle parts, namely sides and angles. The importance of the sine and cosine lies in their ability to be manipulated so as to produce all the other functions. Thus the tangent can be expressed as the ratio of the sine and cosine, that is tangent = sine/cosine. Continuing, we get the cotangent as the reciprocal of the tangent, namely cotangent = 1/tangent. The secant is the reciprocal of the cosine, that is, secant = 1/cosine; and the cosecant is the reciprocal of the sine, that is cosecant = 1/sine.

Since the sine and cosine are so important, it would behoove us to know what they are. Well the answer may startle you because of its simplicity. The sine and cosine are nothing more than the ratio of two sides of a right triangle. Since the sides of a right triangle are assigned a number to represent their lengths, we see that the sine and cosine are nothing more than certain fractions! But what ratios are they? Let us see.

If we take a right triangle and label the vertices A,B, and C such that C is the vertex of the right angle (the angle of measure 90 degrees) then the hypotenuse, the longest side of the triangle is opposite this largest angle. If we take one of the other acute angles at A or B, we express the sine or cosine, abbreviated sin and cos, respectively, in terms of the ratios of the sides of the triangle. These formulas are given by a famous mnemonic SOHCAHTOA. Each of the letters stands for, consecutively, sine, opposite, hypotenuse, cosine, adjacent, hypotenuse, tangent, opposite, adjacent. Thus sine = opposite/hypotenuse, that is, the sine is the ratio of the side opposite the given angle to the hypotenuse. Specifically, if we use vertex A (angle A), then sin(A) is equal to the ratio of the side opposite A to the hypotenuse; similarly, the cos(A) is equal to the ratio of the side adjacent to A (the non-hypotenuse adjacent side) to the hypotenuse. Similarly, the tangent is equal to the side opposite A to the side adjacent to A.

It may seem difficult to believe that out of these basic ideas springs a branch of mathematics that has applications to no less than thirty diverse fields, spanning everything from navigation to medical imaging (CAT Scans) to computer graphics. Yet this is what we get from two seemingly innocent ratios. So the next time you hear trigonometry and start to cringe, remember that this subject springs from the sine and cosine, and these are nothing more than two lonely ratios, looking to have some fun. The rest is all smoke and mirrors.