If you read the first article in this series, then you know the sine is an important trigonometric function. Yet as important as the sine is, and as myriad the formulas and mathematical laws that it finds itself entwined in, one finds in the study of trigonometry an equally formidable ally: the cosine.

The cosine, or as its name implies, the sine's complement, tells us the ratio of the side adjacent the given angle to the hypotenuse. In the oft taught *SOHCAHTOA*, the CAH part stands for *cosine = adjacent/hypotenuse*. This is the mnemonic that most students are taught when they are introduced to the basic trigonometric functions, of which sine, cosine, and tangent are the core.

Like the sine, the cosine is a periodic function. A periodic function is one whose values repeat regularly. For the sine and cosine, their maximum value is 1, their minimum value is -1, and both functions oscillate endlessly between these two values as you move along the real number line in either direction. For this reason, both the sine and cosine serve very well as functional models of any type of physical phenomena with periodic behaviors, i. e. , pendulums, alternating current, oceanic waves, tides, springs, electromagnetic radiation, musical tones, and the car engine.

The famous *Law of Cosines *allows us to solve non-right triangles. That is, if we are given two side lengths of a non-right triangle and the angle between them, we can calculate the measure of the missing two angles and the other side. This is called “solving the triangle. " This would be useful, for instance, if we were building a very large triangular structure, let us say, out of metal or stone. If for purposes of the structure, we were confined to two sides of a particular length and a specific degree measure for the angle between these given sides, we could use the *Law of Cosines *to calculate what the length of the other side would have to be. Architectural feats as these are facilitated by such mathematical laws.

That a simple ratio between the sides of a right triangle would find itself privy to many applications in the real world is not only a statement full of wonder but also a fact of great utility. Do yourself a favor: befriend Chief SOHCAHTOA.

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*February 24, 2008*(582)