The tangent is the last of the three principal trigonometric functions. It derives its name from the Latin tangere for “to touch. " This derivation is relevant to the actual mechanics of the function and the manner in which the tangent works to give us some important measurements in life. You see, the tangent allows us to compute the maximum and minimum values of a function, and this application has significant weight in the real world.
The tangent of a given angle in a right triangle tells us the quotient of the side opposite this angle to the side adjacent to it. In the oft taught SOHCAHTOA, the TOA part stands for tangent = opposite/adjacent. This is the mnemonic that most students are taught when they are introduced to the basic trigonometric functions, of which sine, cosine, and tangent form the core.
Like the sine and cosine, the tangent is also a periodic function. Unlike the sine and cosine, however, the tangent is not defined for certain values along the x-axis, and these values occur at the points which are the odd multiples of pi/2. By undefined is meant that the tangent grows increasingly positive or negative at these values. Mathematicians say that the tangent “grows without bound" here, or that the tangent “approaches infinity" at these values.
The tangent is related to a very important concept in algebra: the slope. If you recall, the slope of a line is defined by the rise over the run, or the change in the y values over the change in the x values. What the slope is measuring is nothing more than the inclination of the given line. Thus a higher value for the slope means that the line is steeper than a line with a smaller slope. If you draw a right triangle in the coordinate plane, with one of the sides parallel to the x-axis, and the hypotenuse with a positive slope (rising from left to right), then the tangent of the acute angle formed by the side parallel to the x-axis and the hypotenuse, is the opposite side over the adjacent side or the change in the y values over the change in the x values. This is precisely the slope of the hypotenuse.
The tangent line to a curve or surface is a line that passes through the curve in only one point, unlike a secant line which passes through the curve or surface in two points. The tangent line thus gently “touches" the surface of the curve and does not cut it. Now where this becomes very important is in the calculus, where the tangent line to a curve is found by calculating the derivative and evaluating at a given point. You see the tangent line can tell us where a given curve reaches both its highest and lowest values. How is this so?
Well if you think about it, picture a curve drawn in the coordinate plane. More specifically, picture a curvy line going from left to right with a number of “hills" and “valleys. " The hills represent where the curve reaches a local high point, and the valleys are the points where the curve reaches a local low point. If the tangent is a line which crosses the curve in one point, all we need to do, to find these highs and lows, is see where the tangent line is horizontal. This is where the curve tops or bottoms out. This is another way of saying we seek to find where the slope of the tangent line is zero, since horizontal lines have zero slopes.
Thus the “touchy" tangent finds itself involved in a very important application in mathematics: that of giving us the maximum and minimum values of a function. Since functions model real life phenomena, this would seem an extremely important thing. For instance, if a function modeled the profits of a major corporation, then knowing the maximum and minimum values would tell us where the profit was highest and where lowest. Having this information just might allow us to tweak the parameters of the profit model to make more or less money. Now wouldn't you think companies might want to know this kind of stuff?
Yes, trigonometry finds itself enmeshed within our world. The three trigonometric functions do more than tell us the ratio of sides of a right triangle, they help tell us about life itself. Keep this in mind next time you run into the sine, cosine, or tangent.
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