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

Maker USS Code 39 in VS .NET f (q) = sin2 q In illustration B, the solid black curve is the graph of g (q) = cos2 q

f (q) = sin2 q In illustration B, the solid black curve is the graph of g (q) = cos2 q
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Figure 15-2
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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
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Each horizontal division is p/2 units
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f (q )
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Each vertical division is 1/4 unit
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Figure 15-3 The dashed gray
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curves are graphs of the sine function (at A) and the cosine function (at B) The solid black curves are graphs of the square of the sine function (at A) and the square of the cosine function (at B) Each division on the horizontal axes represents p /2 units Each division on the vertical axes represents 1/4 unit
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Each horizontal division is p /2 units
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f (q )
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Each vertical division is 1/4 unit Each horizontal division is p /2 units g(q)
Each vertical division is 1/4 unit
288 Trigonometric Curves
The squared-function 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 squared-function waves are equal to 1, while the minimum peak amplitudes are equal to 0 We define the peak-to-peak 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 peak-to-peak amplitudes of 1 ( 1) = 2, while the squared-function waves have peak-to-peak 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 15-4 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 15-4
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 sum-of-squares function is indeed the horizontal, solid black line portrayed in Fig 15-4
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 15-5 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 sine-squared and cosine-squared functions
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