qr code vb.net Figure 15-5 Graph of the ratio of in VS .NET

Painting Code 3 of 9 in VS .NET Figure 15-5 Graph of the ratio of

Figure 15-5 Graph of the ratio of
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the square of the sine function to the square of the cosine function (solid black curves) Each horizontal division represents p /2 units Each vertical division represents 1/2 unit The dashed gray curves are the graphs of the original sinesquared and cosinesquared functions The vertical dashed lines are asymptotes of f
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f (q )
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Each horizontal division is p /2 units
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Each vertical division is 1/2 unit
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This ratio function f is singular (that is, it blows up ) when q is any odd-integer multiple of p /2 That s because cos2 q (the denominator) equals 0 at those points, while sin2 q (the numerator) equals 1 The function attains values of 0 at all integer multiples of p because at those points, sin2 q (the numerator) equals 0, while cos2 q (the denominator) equals 1 The period of f is p, the distance between the asymptotes; the graph repeats itself completely between each adjacent pair of asymptotes The peak amplitude and the peak-to-peak amplitude are both undefined (It s tempting to call them infinite, but let s not go there!) The domain includes all reals except the odd-integer multiples of p /2 The range is the set of all nonnegative reals
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Graphs Involving the Secant and Cosecant
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In Chap 2, we saw graphs of the basic secant and cosecant functions, which are the reciprocals of the cosine and sine, respectively Let s combine these two functions after the fashion of the previous section, and see what the resulting graphs look like
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Secant and cosecant: example 1 The dashed gray curves in Fig 15-6 are the superimposed graphs of the secant and cosecant functions The complex of solid black curves is a graph of their sum As always, you can reproduce this graph by inputting a sufficient number of values into your calculator, plotting the
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Figure 15-6 Graph of the sum of
the secant and cosecant functions (solid black curves) The dashed gray curves are the graphs of the original functions Each division on the horizontal axis represents p/2 units Each vertical division represents 1 unit The vertical dashed lines are asymptotes of f The dependentvariable axis is also an asymptote of f
f(q )
Each horizontal division is p /2 units Each vertical division is 1 unit
Graphs Involving the Secant and Cosecant
output points, and then connecting the dots You ll have to take some time to investigate this function before you can accurately plot this graph, but be patient! We have f (q) = sec q + csc q The graph of f has asymptotes that pass through every point where the independent variable is an integer multiple of p/2 If you examine Fig 15-6 closely, you ll see that the graph is regular and it repeats with a period of 2p, but we can t call it a wave The domain includes all real numbers except the integer multiples of p/2 because, whenever q attains one of those values, either the secant or the cosecant is undefined The range spans the set of all real numbers
Secant and cosecant: example 2 Figure 15-7 shows graphs of the secant and cosecant functions along with their product The dashed gray curves are graphs of the original functions superimposed on each other; the solid black curves show the graph of
f (q) = sec q csc q This function f has a period of p, which is half that of the secant and cosecant functions Like the sum-function graph, this graph has asymptotes that pass through every point where the independent variable is an integer multiple of p /2 The domain is the set of all reals except the integer multiples of p /2 The range spans the set of all real numbers except those in the open interval ( 2,2) Alternatively, we can say that the range includes all reals y such that y 2 or y 2 Figure 15-7 Graph of the
product of the secant and cosecant functions (solid black curves) The dashed gray curves are the graphs of the original functions Each division on the horizontal axis represents p/2 units Each vertical division represents 1 unit The vertical dashed lines are asymptotes of f The dependentvariable axis is also an asymptote of f
f(q )
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