| Trigonometry is hailed as a rough subject, particularly | | | | assigned a number to represent their lengths, we see |
| for precocious high school students brave enough to | | | | that the sine and cosine are nothing more than certain |
| wade into its potentially turbulent waters. | | | | fractions! But what ratios are they? Let us see. |
| Notwithstanding "trig" challenges, the key to mastering | | | | If we take a right triangle and label the vertices A,B, |
| this subject really lies in the basic understanding of its | | | | and C such that C is the vertex of the right angle (the |
| two key players: the sine and cosine. All the other | | | | angle of measure 90 degrees) then the hypotenuse, |
| features of this fascinating subject really derive from | | | | the longest side of the triangle is opposite this largest |
| the shenanigans that these two play on unsuspecting | | | | angle. If we take one of the other acute angles at A |
| students. Consequently, before students plunge into this | | | | or B, we express the sine or cosine, abbreviated sin |
| mysterious realm of triangular measurement and other | | | | and cos, respectively, in terms of the ratios of the |
| identities, they should make sure they are intimately | | | | sides of the triangle. These formulas are given by a |
| familiar with its two key players and the critical roles | | | | famous mnemonic SOHCAHTOA. Each of the letters |
| they play. | | | | stands for, consecutively, sine, opposite, hypotenuse, |
| Trigonometry derives from the Greek "trigonon" = | | | | cosine, adjacent, hypotenuse, tangent, opposite, |
| triangle and "metron" = to measure. In essence, | | | | adjacent. Thus sine = opposite/hypotenuse, that is, the |
| trigonometry means the measure of triangles, and as | | | | sine is the ratio of the side opposite the given angle to |
| such all the trigonometric functions derive from the | | | | the hypotenuse. Specifically, if we use vertex A (angle |
| triangle, specifically the right triangle; however, the true | | | | A), then sin(A) is equal to the ratio of the side opposite |
| science encompasses a lot more than the simple | | | | A to the hypotenuse; similarly, the cos(A) is equal to |
| measurement of triangle parts, namely sides and | | | | the ratio of the side adjacent to A (the |
| angles. The importance of the sine and cosine lies in | | | | non-hypotenuse adjacent side) to the hypotenuse. |
| their ability to be manipulated so as to produce all the | | | | Similarly, the tangent is equal to the side opposite A to |
| other functions. Thus the tangent can be expressed | | | | the side adjacent to A. |
| as the ratio of the sine and cosine, that is tangent = | | | | It may seem difficult to believe that out of these basic |
| sine/cosine. Continuing, we get the cotangent as the | | | | ideas springs a branch of mathematics that has |
| reciprocal of the tangent, namely cotangent = 1 | | | | applications to no less than thirty diverse fields, |
| tangent. The secant is the reciprocal of the cosine, that | | | | spanning everything from navigation to medical imaging |
| is, secant = 1/cosine; and the cosecant is the reciprocal | | | | (CAT Scans) to computer graphics. Yet this is what |
| of the sine, that is cosecant = 1/sine. | | | | we get from two seemingly innocent ratios. So the |
| Since the sine and cosine are so important, it would | | | | next time you hear trigonometry and start to cringe, |
| behoove us to know what they are. Well the answer | | | | remember that this subject springs from the sine and |
| may startle you because of its simplicity. The sine and | | | | cosine, and these are nothing more than two lonely |
| cosine are nothing more than the ratio of two sides of | | | | ratios, looking to have some fun. The rest is all smoke |
| a right triangle. Since the sides of a right triangle are | | | | and mirrors. |