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print barcode image c# Basics in Software
Basics Draw QR Code JIS X 0510 In None Using Barcode printer for Software Control to generate, create QR Code ISO/IEC18004 image in Software applications. QR Decoder In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. 2F/3 Encoding Quick Response Code In Visual C# Using Barcode generation for .NET Control to generate, create QR Code image in .NET applications. QR Code Printer In .NET Using Barcode creation for ASP.NET Control to generate, create Denso QR Bar Code image in ASP.NET applications. Fig 129
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Barcode Encoder In None Using Barcode creation for Software Control to generate, create bar code image in Software applications. Code 3 Of 9 Generation In None Using Barcode maker for Software Control to generate, create Code 39 image in Software applications. Several angles are sketched in Fig 131, and both their radian and degree measures given
Encode UCC.EAN  128 In None Using Barcode generation for Software Control to generate, create EAN128 image in Software applications. Print Bar Code In None Using Barcode printer for Software Control to generate, create barcode image in Software applications. If is an angle, let (x, y) be the coordinates of the terminal point of the corresponding radius (called the terminal radius) on the unit circle We call P = (x, y) the terminal point corresponding to Look at Fig 132 The number y is called the sine of and is written sin The number x is called the cosine of and is written cos Since (cos , sin ) are coordinates of a point on the unit circle, the following two fundamental properties are immediate: (1) For any number , (sin )2 + (cos )2 = 1 Draw GTIN  8 In None Using Barcode creator for Software Control to generate, create EAN8 image in Software applications. UCC  12 Generator In None Using Barcode creator for Online Control to generate, create UPCA Supplement 5 image in Online applications. CHAPTER 1 Basics
GS1128 Creator In Visual Basic .NET Using Barcode generation for .NET Control to generate, create EAN128 image in .NET framework applications. Matrix 2D Barcode Creator In Java Using Barcode drawer for Java Control to generate, create Matrix 2D Barcode image in Java applications. 5F/6 = 150 EAN / UCC  13 Drawer In VS .NET Using Barcode creation for ASP.NET Control to generate, create UCC  12 image in ASP.NET applications. Recognize UPC A In Java Using Barcode reader for Java Control to read, scan read, scan image in Java applications. = 60 Encoding GS1 128 In Java Using Barcode creator for Java Control to generate, create GS1128 image in Java applications. Drawing Code 3 Of 9 In Java Using Barcode creation for Java Control to generate, create Code 3/9 image in Java applications. _ p = _180 x _ 3p/4 = _135 Fig 131
unit circle
P = (x, y) G cos G
sin G x
Fig 132
For any number , 1 cos 1 and 1 sin 1
Math Note: It is common to write sin2 to mean (sin )2 and cos2 to mean (cos )2 EXAMPLE 118
Compute the sine and cosine of /3 Basics
SOLUTION We sketch the terminal radius and associated triangle (see Fig 133) This is a 30 60 90 triangle whose sides have ratios 1 : 3 : 2 Thus 1 =2 x Likewise, y = 3 x It follows that or y = 3x = or 1 x= 2 3 2 3 sin = 3 2 1 = 3 2
and cos
unit circle 3 2 x
Fig 133
You Try It: The cosine of a certain angle is 2/3 The angle lies in the fourth quadrant What is the sine of the angle Math Note: Notice that if is an angle then and + 2 have the same terminal radius and the same terminal point (for adding 2 just adds one more trip around the circle look at Fig 134) As a result, sin = x = sin( + 2 ) CHAPTER 1 Basics
G cos G
sin G x
unit circle
Fig 134
and cos = y = cos( + 2 ) We say that the sine and cosine functions have period 2 : the functions repeat themselves every 2 units In practice, when we calculate the trigonometric functions of an angle , we reduce it by multiples of 2 so that we can consider an equivalent angle , called the associated principal angle, satisfying 0 < 2 For instance, 15 /2 has associated principal angle 3 /2 (since 15 /2 3 /2 = 3 2 ) and 10 /3 2 /3 has associated principal angle (since 10 /3 2 /3 = 12 /3 = 2 2 ) You Try It: What are the principal angles associated with 7 , 11 /2, 8 /3, 14 /5, 16 /7 What does the concept of angle and sine and cosine that we have presented here have to do with the classical notion using triangles Notice that any angle such that 0 < /2 has associated to it a right triangle in the rst quadrant, with vertex on the unit circle, such that the base is the segment connecting (0, 0) to (x, 0) and the height is the segment connecting (x, 0) to (x, y) See Fig 135 y opposite side
G adjacent side
Basics
unit circle
(x, y) Fig 135
Then sin = y = and cos = x = adjacent side of triangle x = 1 hypotenuse opposite side of triangle y = 1 hypotenuse Thus, for angles between 0 and /2, the new de nition of sine and cosine using the unit circle is clearly equivalent to the classical de nition using adjacent and opposite sides and the hypotenuse For other angles , the classical approach is to reduce to this special case by subtracting multiples of /2 Our approach using the unit circle is considerably clearer because it makes the signatures of sine and cosine obvious Besides sine and cosine, there are four other trigonometric functions: y x x cot = y 1 sec = x 1 csc = y tan = sin cos cos = sin 1 = cos 1 = sin = Whereas sine and cosine have domain the entire real line, we notice that tan and sec are unde ned at odd multiples of /2 (because cosine will vanish there) and cot and csc are unde ned at even multiples of /2 (because sine will vanish there) The graphs of the six trigonometric functions are shown in Fig 136 EXAMPLE 119 Compute all the trigonometric functions for the angle = 11 /4

