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qr code vb.net open source y x = a cos f in Visual Studio .NET
y x = a cos f Code39 Reader In Visual Studio .NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in .NET framework applications. Make Code 39 In .NET Using Barcode printer for .NET framework Control to generate, create Code 39 image in .NET applications. (x, y) y = a sin f
Code 39 Decoder In .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in VS .NET applications. Bar Code Generation In VS .NET Using Barcode creation for Visual Studio .NET Control to generate, create barcode image in .NET applications. Figure 1610 Barcode Decoder In Visual Studio .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET applications. Draw Code39 In Visual C# Using Barcode drawer for .NET framework Control to generate, create Code 39 Full ASCII image in Visual Studio .NET applications. Cartesiancoordinate graph of a circle with radius a, centered at the origin We can let f = t to describe this circle as a pair of parametric equations USS Code 39 Printer In VS .NET Using Barcode printer for ASP.NET Control to generate, create Code39 image in ASP.NET applications. Code39 Drawer In Visual Basic .NET Using Barcode generator for .NET framework Control to generate, create Code 39 Full ASCII image in VS .NET applications. Parametric Equations in TwoSpace
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Does the above pair of parametric equations seem strange to you The second equation doesn t contain the parameter! That s not a problem in this situation The parameter has no effect because the polar radius r is always the same Here s a challenge! Suppose that we come across a pair of parametric equations similar to the one in the Cartesiancoordinate example above, except that the cosine and sine of the parameter are multiplied by different nonzero realnumber constants a and b, like this: x = a cos t From Graph to Equations
and y = b sin t What sort of curve should we expect to get if we graph the relation defined by this pair of parametric equations Solution
We ve been told that a and b are both nonzero real numbers Therefore, we can divide the equations through by their respective constants to get x /a = cos t and y /b = sin t If we square both sides of both equations, we obtain (x /a)2 = cos2 t and (y /b)2 = sin2 t When we add these two equations, lefttoleft and righttoright, we obtain the new equation (x /a)2 + (y /b)2 = cos2 t + sin2 t From trigonometry, we remember that for any real number t, it s always true that cos2 t + sin2 t = 1 Therefore, the preceding equation can be rewritten as (x /a)2 + (y /b)2 = 1 Expanding the squared ratios on the lefthand side gives us x2 /a2 + y2/b2 = 1 which is the equation of an ellipse centered at the origin The horizontal (xcoordinate) semiaxis is a units wide, and the vertical (ycoordinate) semiaxis is b units high Parametric Equations in TwoSpace
Practice Exercises
This is an openbook quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find workedout answers in App B The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 In Rectangularcoordinate graph: example 1 (Fig 164), the parametric equations are x = t2 and y = t3 Find an equation for this relation that expresses x in terms of y without the parameter t Then find an equation that expresses y in terms of x without the parameter t 2 Is the relation defined in your second answer to Problem 1 a function of x 3 In Polarcoordinate graph: example 2 (Fig 167), the parametric equations are q = t 1 and r = ln t Find an equation for this relation that expresses q in terms of r without the parameter t Then find an equation that expresses r in terms of q without the parameter t 4 Is the relation defined in your second answer to Problem 3 a function of q 5 In Cartesiancoordinate graph to equations (Fig 1610), the parametric equations are x = a cos t and y = a sin t where a is a nonzero constant Find an equation for this relation in terms of x and y only, without the parameter t 6 Express the solution to Problem 5 as a relation in which x is the independent variable and y is the dependent variable You should end up with y alone on the lefthand side of the equals sign, and an expression containing x (but not y) on the righthand side Is this relation a function of x 7 Suppose that we come across the pair of parametric equations x = sec t and y = tan t Find an equation for this relation in terms of x and y only, without the parameter t What sort of curve should we expect to get if we graph this relation in the Cartesian xy plane

