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multivariable_04_Transcendental_Functions

multivariable_04_Transcendental_Functions - 4...

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4 Transcendental Functions So far we have used only algebraic functions as examples when finding derivatives, that is, functions that can be built up by the usual algebraic operations of addition, subtraction, multiplication, division, and raising to constant powers. Both in theory and practice there are other functions, called transcendental, that are very useful. Most important among these are the trigonometric functions, the inverse trigonometric functions, exponential functions, and logarithms. When you first encountered the trigonometric functions it was probably in the context of “triangle trigonometry,” defining, for example, the sine of an angle as the “side opposite over the hypotenuse.” While this will still be useful in an informal way, we need to use a more expansive definition of the trigonometric functions. First an important note: while degree measure of angles is sometimes convenient because it is so familiar, it turns out to be ill-suited to mathematical calculation, so (almost) everything we do will be in terms of radian measure of angles. 59

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60 Chapter 4 Transcendental Functions To define the radian measurement system, we consider the unit circle in the xy -plane: . . . x (cos x, sin x ) y A B (1 , 0) An angle, x , at the center of the circle is associated with an arc of the circle which is said to subtend the angle. In the figure, this arc is the portion of the circle from point (1 , 0) to point A . The length of this arc is the radian measure of the angle x ; the fact that the radian measure is an actual geometric length is largely responsible for the usefulness of radian measure. The circumference of the unit circle is 2 πr = 2 π (1) = 2 π , so the radian measure of the full circular angle (that is, of the 360 degree angle) is 2 π . While an angle with a particular measure can appear anywhere around the circle, we need a fixed, conventional location so that we can use the coordinate system to define properties of the angle. The standard convention is to place the starting radius for the angle on the positive x -axis, and to measure positive angles counterclockwise around the circle. In the figure, x is the standard location of the angle π/ 6, that is, the length of the arc from (1 , 0) to A is π/ 6. The angle y in the picture is π/ 6, because the distance from (1 , 0) to B along the circle is also π/ 6, but in a clockwise direction. Now the fundamental trigonometric definitions are: the cosine of x and the sine of x are the first and second coordinates of the point A , as indicated in the figure. The angle x shown can be viewed as an angle of a right triangle, meaning the usual triangle definitions of the sine and cosine also make sense. Since the hypotenuse of the triangle is 1, the “side opposite over hypotenuse” definition of the sine is the second coordinate of point A over 1, which is just the second coordinate; in other words, both methods give the same value for the sine.
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