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Trigonometric Functions of Quadrantal Angles in Visual Studio .NET
Trigonometric Functions of Quadrantal Angles QR Code ISO/IEC18004 Recognizer In .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in VS .NET applications. Making QR Code In VS .NET Using Barcode generation for .NET Control to generate, create QR Code 2d barcode image in .NET applications. For a quadrantal angle, the terminal side coincides with one of the axes. A point P, distinct from the origin, on the terminal side has either x 0 and y 0, or x 0 and y 0. In either case, two of the six functions will not be defined. For example, the terminal side of the angle 0 coincides with the positive x axis and the ycoordinate of P is 0. Since the xcoordinate occurs in the denominator of the ratio defining the cotangent and cosecant, these functions are not defined. In this book, undefined will be used instead of a numerical value in such cases, but some authors indicate this by writing cot 0 , and others write cot 0 . The following results are obtained in Prob. 2.17. Quick Response Code Recognizer In Visual Studio .NET Using Barcode decoder for .NET framework Control to read, scan read, scan image in .NET applications. Bar Code Creation In .NET Framework Using Barcode creator for VS .NET Control to generate, create bar code image in .NET framework applications. Angle 0 90 180 270 sin 0 1 0 1 cos 1 0 1 0 tan 0 Undefined 0 Undefined cot Undefined 0 Undefined 0 sec 1 Undefined 1 Undefined csc Undefined 1 Undefined 1 Read Bar Code In Visual Studio .NET Using Barcode decoder for .NET Control to read, scan read, scan image in .NET framework applications. Making QR Code ISO/IEC18004 In Visual C# Using Barcode maker for VS .NET Control to generate, create QR Code 2d barcode image in Visual Studio .NET applications. 2.7 Undefined Trigonometric Functions
Printing QR Code ISO/IEC18004 In .NET Using Barcode drawer for ASP.NET Control to generate, create QR Code image in ASP.NET applications. Encode QR Code In Visual Basic .NET Using Barcode maker for .NET Control to generate, create QR image in .NET framework applications. It has been noted that cot 0 and csc 0 are not defined since division by zero is never allowed, but the values of these functions for angles near 0 are of interest. In Fig. 2.7(a), take u to be a small positive angle in standard position and on its terminal side take P(x, y) to be at a distance r from O. Now x is slightly less than r, 2D Barcode Creation In .NET Using Barcode drawer for .NET framework Control to generate, create Matrix 2D Barcode image in VS .NET applications. Make EAN13 In Visual Studio .NET Using Barcode creator for .NET Control to generate, create EAN13 image in Visual Studio .NET applications. CHAPTER 2 Trigonometric Functions of a General Angle
Create Bar Code In .NET Framework Using Barcode encoder for Visual Studio .NET Control to generate, create barcode image in .NET applications. USD  8 Generator In Visual Studio .NET Using Barcode encoder for VS .NET Control to generate, create Code 11 image in .NET framework applications. and y is positive and very small; then cot x/y and csc r/y are positive and very large. Next let decrease toward 0 with P remaining at a distance r from O. Now x increases but is always less than r, while y decreases but remains greater than 0; thus cot and csc become larger and larger. (To see this, take r 1 and compute csc when y 0.1, 0.01, 0.001, . . . .) This state of affairs is indicated by If approaches 0 , then cot approaches , which is what is meant when writing cot 0 . Code 128 Code Set A Creator In ObjectiveC Using Barcode printer for iPhone Control to generate, create Code 128 Code Set A image in iPhone applications. Encoding USS128 In ObjectiveC Using Barcode generation for iPad Control to generate, create EAN128 image in iPad applications. Fig.2.7
GTIN  13 Creation In Java Using Barcode creator for Java Control to generate, create EAN 13 image in Java applications. Code39 Drawer In None Using Barcode creator for Software Control to generate, create Code39 image in Software applications. Next suppose, as in Fig. 2.7(b), that is a negative angle close to 0 , and take P(x, y) on its terminal side at a distance r from O. Then x is positive and slightly smaller than r, while y is negative and has a small absolute value. Both cot and csc are negative with large absolute values. Next let increase toward 0 with P remaining at a distance r from O. Now x increases but is always less than r, while y remains negative with an absolute value decreasing toward 0; thus cot and csc remain negative, but have absolute values that get larger and larger. This situation is indicated by If approaches 0 , then cot approaches , which is what is meant when writing cot 0 . In each of these cases, cot 0 and cot 0 , the use of the sign does not have the standard meaning of equals and should be used with caution, since cot 0 is undefined and is not a number. The notation is used as a short way to describe a special situation for trigonometric functions. The behavior of other trigonometric functions that become undefined can be explored in a similar manner. The following chart summarizes the behavior of each trigonometric function that becomes undefined for angles from 0 up to 360 . 1D Barcode Creation In Java Using Barcode encoder for Java Control to generate, create Linear 1D Barcode image in Java applications. Encoding UPCA In Java Using Barcode generation for Android Control to generate, create UPCA Supplement 5 image in Android applications. Angle u Function Values 0 0
GTIN  12 Creation In Visual C# Using Barcode creator for Visual Studio .NET Control to generate, create GS1  12 image in .NET framework applications. Creating Barcode In None Using Barcode generator for Excel Control to generate, create bar code image in Microsoft Excel applications. u u
cot cot tan tan cot cot tan tan
and csc and csc
u 90 u 90
and sec and sec and csc
u 180 u 270 u 270
u 180 and csc and sec and sec
(NOTE: The means the value is greater than the number stated; 180 means values greater than 180 . The means the value is less than the number stated; 90 means values less than 90 .) 2.8 Coordinates of Points on a Unit Circle
Let s be the length of an arc on a unit circle x2 y2 1; each s is paired with an angle in radians (see Sec. 1.4). Using the point (1, 0) as the initial point of the arc and P(x, y) as the terminal point of the arc, as in Fig. 2.8, we can determine the coordinates of P in terms of the real number s.

