Circular trigonometric functions

The ordinary trigonometric functions include the sine, the cosine, the tangent, the cotangent, the secant, and the cosecant. These are sometimes called the circular trigonometric functions to distinguish them from the hyperbolic trigonometric functions discussed briefly in the next section of this chapter. 

Find the value of the hyperbolic sine, cosine, and tangent for jc = 0 and x = n/2. Compare these values with the values of the ordinary (circular) trigonometric functions for the same values of the independent variable. 

The trigonometric functions developed in the previous seefidtiare lef pd to as circidar functions, as they are related to the circle shown in Fig, 11, somewhat less familiar family of functions, the hyperbolic funefens, c also be derived from the exponential. They are analogous to the circular iom considered above and can be defined bv the relations... 

Figure 6.14 The displacement modes of a circular rod. The trigonometric functions are plotted using reference circles as zero amplitude. Positive displacements are on the outside of the reference circles.

This is called the circular measure of an angle and, for this reason, trigonometrical functions are sometimes called circular functions. This property is possessed by no plane curve other than the circle. For instance, the hyperbola, though symmetrically placed with respect to its centre, is not at all points equidistant from it. The same thing is true of the ellipse. The parabola has no centre. 

Figure 2.10 shows the hyperbolic sine and hyperbolic cosine for values of x from 0 to 3. Note that the values of the hyperbolic sin and the hyperbolic cosine do not necessarily lie between -1 and 1 as do the values of the circular sine and cosine functions and that both functions approach e /2 for large values of x. The hyperbolic trigonometric functions are available in Excel. 

These functions are called circular or trigonometric functions. Note that Equations (2-25) are just the transformation Equations (1-4) with r = 1. It is interesting to compare the graphs of functions, such as sin 0 and cos 0, in linear coordinates (coordinates in which 0 is plotted along one axis) to those in plane polar coordinates. Consider, for example, the equation r = A cos 0y where A is a constant. Such an equation can be used to describe the wave properties of p-type atomic orbitals in two dimensions. The functional dependence of r upon 0 can be seen in Table 2-1. 

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