Arccos b 3 Arccot 15 b 8 in VS .NET

Make QR Code JIS X 0510 in VS .NET Arccos b 3 Arccot 15 b 8

1 Arccos b 3 Arccot 15 b 8
Reading QR Code ISO/IEC18004 In VS .NET
Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in .NET applications.
QR Printer In VS .NET
Using Barcode creator for .NET Control to generate, create Quick Response Code image in VS .NET applications.
(g) tan a2 Arcsin (h) sin a2 Arcsin
Decoding QR Code JIS X 0510 In Visual Studio .NET
Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in .NET framework applications.
Encoding Bar Code In Visual Studio .NET
Using Barcode creator for .NET Control to generate, create barcode image in Visual Studio .NET applications.
13.23 Prove: The area of the segment cut from a circle of radius r by a chord at a distance d from the center is given by K r2 Arccos d>r d2r2 d2.
Recognizing Bar Code In .NET Framework
Using Barcode reader for .NET framework Control to read, scan read, scan image in VS .NET applications.
Creating QR Code 2d Barcode In C#
Using Barcode maker for .NET framework Control to generate, create QR-Code image in .NET applications.
CHAPTER 12 14
Draw QR In .NET Framework
Using Barcode generator for ASP.NET Control to generate, create QR image in ASP.NET applications.
QR Code Printer In Visual Basic .NET
Using Barcode drawer for VS .NET Control to generate, create QR Code image in Visual Studio .NET applications.
Trigonometric Equations
Linear Barcode Printer In .NET Framework
Using Barcode maker for Visual Studio .NET Control to generate, create 1D Barcode image in Visual Studio .NET applications.
Encode EAN13 In .NET Framework
Using Barcode maker for Visual Studio .NET Control to generate, create EAN13 image in VS .NET applications.
14.1 Trigonometric Equations
Barcode Drawer In .NET
Using Barcode printer for VS .NET Control to generate, create bar code image in Visual Studio .NET applications.
Drawing MSI Plessey In VS .NET
Using Barcode creator for .NET framework Control to generate, create MSI Plessey image in VS .NET applications.
Trigonometric equations, i.e., equations involving trigonometric functions of unknown angles, are called: (a) Identical equations, or identities, if they are satisfied by all values of the unknown angles for which the functions are defined (b) Conditional equations, or simply equations, if they are satisfied only by particular values of the unknown angles
Read UPC A In VB.NET
Using Barcode recognizer for .NET framework Control to read, scan read, scan image in .NET framework applications.
Drawing Data Matrix In Java
Using Barcode generation for Java Control to generate, create Data Matrix image in Java applications.
EXAMPLE 14.1 (a) sin x csc x
Scanning Code 39 In VB.NET
Using Barcode scanner for .NET Control to read, scan read, scan image in Visual Studio .NET applications.
Barcode Generation In None
Using Barcode generation for Microsoft Word Control to generate, create bar code image in Microsoft Word applications.
1 is an identity, being satisfied by every value of x for which csc x is defined.
Encode UPC Code In Java
Using Barcode generator for Java Control to generate, create UPC-A Supplement 2 image in Java applications.
Barcode Creation In .NET Framework
Using Barcode generation for ASP.NET Control to generate, create bar code image in ASP.NET applications.
1 4p
Painting Code 39 Full ASCII In None
Using Barcode drawer for Font Control to generate, create Code 39 image in Font applications.
Generate GTIN - 128 In None
Using Barcode creation for Microsoft Excel Control to generate, create EAN / UCC - 14 image in Microsoft Excel applications.
(b) sin x
0 is a conditional equation since it is not satisfied by x
or 1p. 2
Hereafter in this chapter we will use the term equation instead of conditional equation. A solution of a trigonometric equation like sin x 0 is a value of the angle x which satisfies the equation. Two solutions of sin x 0 are x 0 and x . If a given equation has one solution, it has in general an unlimited number of solutions. Thus, the complete solution of sin x 0 is given by x 0 2n x 2n x<2 .
where n is any positive or negative integer or zero. In this chapter we will list only the particular solutions for which 0
14.2 Solving Trigonometric Equations
There is no general method for solving trigonometric equations. Several standard procedures are illustrated in the following examples, and other procedures are introduced in the Solved Problems. All solutions will be for the interval 0 x < 2 . (A) The equation may be factorable.
EXAMPLE 14.2 Solve sin x
2 sin x cos x
0. sin x 2 sin x cos x 0
Factor: sin x (1
2 cos x)
0 0 or 1 or cos x x 2 cos x 1/2 /3, 5 /3 0
Set each factor equal to 0: sin x Solve each equation for x: x 0,
CHAPTER 14 Trigonometric Equations
Check:
For x For x For x For x
0, sin x /3, sin x , sin x
2 sin x cos x 2 sin x cos x 2 sin x cos x
2(0)(1)
1 2 23 1
2 A 2 23 B A 2 B 0
1 2 23
2(0)( 1)
1 2 23
5 /3, sin x
2 sin x cos x x < 2 ) are x
B A1 B 2
Thus, the required solutions (0
0, /3, , and 5 /3.
(B) The various functions occurring in the equation may be expressed in terms of a single function.
EXAMPLE 14.3 Solve 2 tan2 x
sec2 x
Replacing sec x by 1
tan x, we have (1 tan2 x) 2 3 tan2 x 1 and tan x 1/ 23
2 tan2 x From tan x 1/1 23, x
/6 and 7 /6; from tan x
1/ 23, x
5p/6 and 11 /6. After checking each of these x < 2 ) are x /6, 5 /6, 7 /6, and 11 /6.
values in the original equation, we find that the required solutions (0
The necessity of the check is illustrated in Examples 14.4 and 14.5.
EXAMPLE 14.4 Solve sec x
tan x
0. tan x 1;
Multiplying the equation sec x then x
sin x 1 0 by cos x, we have 1 sin x 0 or sin x cos x cos x 3 /2. However, neither sec x nor tan x is defined when x 3 /2 and the equation has no solution.
(C) Both members of the equation are squared.
EXAMPLE 14.5 Solve sin x
cos x
21 sin 2 x and If the procedure of (B) were used, we would replace sin x by 21 cos 2 x or cos x by thereby introduce radicals. To avoid this, we write the equation in the form sin x 1 cos x and square both members. We have sin2 x 1 2 cos2 x From cos x Check: 0, x 0, /2, 3 /2, /2, 3 /2; from cos x sin x sin x sin x cos x cos x cos x 0 and /2. cos2 x 2 cos x 1, x 0 1 1 1 0 0 0. 1 1 1 1 1 2 cos x 2 cos x cos2 x cos2 x 1) 0 (1)
2 cos x (cos x
For x For x For x
Thus, the required solutions are x
The value x 3 /2, called an extraneous solution, was introduced by squaring the two members. Note that (1) is also obtained when both members of sin x cos x 1 are squared and that x 3 /2 satisfies this latter relation. (D) Solutions are approximate values. (NOTE: Since we will be using real number properties in solving the equation, the approximate values for the angles will be stated in radians, which can be found using Table 3 in Appendix 2 or a calculator. These values are not exact and may not yield an exact check when substituted into the given equation.)
Copyright © OnBarcode.com . All rights reserved.