Trigonometric Functions of Quadrantal Angles in Visual Studio .NET

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Trigonometric Functions of Quadrantal Angles
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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 y-coordinate of P is 0. Since the x-coordinate 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.
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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
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2.7 Undefined Trigonometric Functions
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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,
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CHAPTER 2 Trigonometric Functions of a General Angle
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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 .
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Fig.2.7
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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 .
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Angle u Function Values 0 0
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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.
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