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vb.net generate qr code f (q) = sin2 q In illustration B, the solid black curve is the graph of g (q) = cos2 q in VS .NET
f (q) = sin2 q In illustration B, the solid black curve is the graph of g (q) = cos2 q Code 3 Of 9 Scanner In VS .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET applications. Code 3/9 Generator In .NET Framework Using Barcode encoder for VS .NET Control to generate, create Code39 image in .NET applications. Figure 152 Code 3 Of 9 Scanner In .NET Framework Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in .NET applications. Make Bar Code In .NET Using Barcode generation for .NET framework Control to generate, create barcode image in VS .NET applications. Graphs of the sine and cosine functions (dashed gray curves) along with the graph of their product (solid black curve) Each horizontal division represents p/2 units Each vertical division represents 1/4 unit Bar Code Decoder In Visual Studio .NET Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications. Code 39 Full ASCII Generation In C# Using Barcode maker for .NET framework Control to generate, create Code 39 Extended image in Visual Studio .NET applications. Each horizontal division is p/2 units
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288 Trigonometric Curves
The squaredfunction waves have periods of p, half the periods of the original functions Therefore, the frequencies of the squared functions are twice those of the original functions The waves for the squared functions are displaced upward relative to the waves for the original functions The squared functions attain repeated minima of 0 and repeated maxima of 1 In other words, the positive peak amplitudes of the squaredfunction waves are equal to 1, while the minimum peak amplitudes are equal to 0 We define the peaktopeak amplitude of a regular, repeating wave as the difference between the positive peak value and the negative peak value In this example, the original waves have peaktopeak amplitudes of 1 ( 1) = 2, while the squaredfunction waves have peaktopeak amplitudes of 1 0 = 1 The waves representing the squared functions f and g look like sinusoids, but we can t be certain about that on the basis of their appearance alone The domains of f and g include all real numbers The ranges of f and g are confined to the set of all reals in the closed interval [0,1] Sine and cosine: example 4 Let s add the squared functions from the previous example and graph the result The solid black line in Fig 154 is a graph of the sum of the squares of the sine and the cosine functions, which are shown as superimposed dashed gray curves We have f (q) = sin2 q + cos2 q In this case, the function has a constant value The domain includes all of the real numbers The range is the set containing the single real number 1 Figure 154 Graph of the sum of the squares of the sine and cosine functions (solid black line) The dashed gray curves are the graphs of the original squared functions Each horizontal division represents p/2 units Each vertical division represents 1/4 unit f (q ) Each horizontal division is p /2 units
Each vertical division is 1/4 unit
Graphs Involving the Sine and Cosine
Are you confused
You might wonder, How can we be certain that the graph in the previous example is actually a straight, horizontal line When I input values into my calculator, I always get an output of 1, but I ve learned that even a million examples can t prove a general truth in mathematics Your skepticism shows that you re thinking! But let s remember one of the basic trigonometric identities that we learned in Chap 2 For all real numbers q, the following equation holds true: sin2 q + cos2 q = 1 This fact assures us that in the previous example, we have f (q) = 1 so the graph of the sumofsquares function is indeed the horizontal, solid black line portrayed in Fig 154 Here s a challenge! Sketch a graph of the ratio of the square of the sine function to the square of the cosine function That is, graph f (q) = (sin2 q)/(cos2 q) Solution
The solid black complex of curves in Fig 155 is the graph of the ratio of the square of the sine to the square of the cosine The superimposed gray curves are graphs of the original sinesquared and cosinesquared functions

