# qr code vb.net Each horizontal division is p /2 units Each vertical division is 1 unit in .NET Creator Code 39 Extended in .NET Each horizontal division is p /2 units Each vertical division is 1 unit

Each horizontal division is p /2 units Each vertical division is 1 unit
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Figure 15-12
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Graph of the product of the tangent and cotangent functions (solid black curve) 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
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298 Trigonometric Curves
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The graph of f is a horizontal, straight line with infinitely many holes, with each hole located at a point where q is an integer multiple of p /2 If we want to get creative with our terminology, we can say that the graph of f consists of infinitely many open-ended line segments, each of length p /2, placed end-to-end in a collinear arrangement The domain of f spans the set of all reals except the integer multiples of p /2 The range is the set containing the single real number 1
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Tangent and cotangent: example 3 The dashed gray curves in Fig 15-13 are the graphs of the tangent function (at A) and the cotangent function (at B) The solid black curves in drawing A compose the graph of
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f (q) = tan2 q Figure 15-13
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The solid black curves are graphs of the squares of the tangent function (at A) and the cotangent function (at B) The dashed gray curves are graphs of the original functions Each division on the horizontal axes represents p /2 units Each division on the vertical axes represents 1 unit The vertical dashed lines are asymptotes of f and g At B, the positive dependentvariable axis is also an asymptote of g
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f (q )
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Each horizontal division is p /2 units Each vertical division is 1 unit g(q )
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Graphs Involving the Tangent and Cotangent
The solid black curves in drawing B compose the graph of g (q) = cot2 q Both f and g have periods of p, the same as the periods of the tangent and cotangent functions Therefore, the frequencies of the squared functions are the same as those of the originals Singularities occur in f and g at the same points on the independent-variable axis as they do for the original functions The domain of f is the set of all reals except odd-integer multiples of p /2 The domain of g is the set of all reals except integer multiples of p The ranges of both f and g span the set of nonnegative real numbers
Tangent and cotangent: example 4 Figure 15-14 is a graph of the sum of the tangent-squared function and the cotangent-squared function The solid black curves compose the graph of
f (q) = tan2 q + cot2 q The dashed gray curves are superimposed graphs of the original squared functions This sum function f has a period equal to half that of the original functions, or p /2 The domain includes all reals except the integer multiples of p /2 The range is the set of reals y such that y 2 Figure 15-14 Graph of the sum
of the squares of the tangent and cotangent functions (solid black curves) The dashed gray curves are the graphs of the original squared 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 positive dependentvariable axis is also an asymptote of f
f(q )
Each horizontal division is p /2 units
Each vertical division is 1 unit
300 Trigonometric Curves
Are you confused
You might wonder how we can be sure that the range of the sum-of-squares function graphed in Fig 5-14 is the set of all reals greater than or equal to 2 To understand this, we can use the same reasoning as we did when we added the squares of the secant and the cosecant functions All the minima on the solid black curves in Fig 15-14 correspond to dependent-variable values of 2, because they occur where the graphs of the dashed gray curves intersect At all such points, the tangent squared and cotangent squared are both 1, so their sum is 2 If you move slightly on either side of any such point, the value of the sum-of-squares function increases
Here s a challenge!
Sketch a graph of the ratio of the square of the tangent function to the square of the cotangent function That is, graph f (q) = (tan2 q)/(cot2 q) State the domain and range of f Be careful! There are some tricky restrictions in the domain