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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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Figure B-22 Illustration for the solution to Problem 1
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in Chap 15 Each horizontal division represents p /2 units Each vertical division represents 1/2 unit
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Figure B-23
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Illustration for the solution to Problem 2 in Chap 15 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 Each vertical division is 1/4 unit
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3 The solid black complex of curves in Fig B-24 is the graph of the ratio of the square of the cosine to the square of the sine If we call the function h, then h (q) = (cos2 q)/(sin2 q) The superimposed gray curves are graphs of the original sine-squared and cosinesquared functions The domain of h includes all real numbers except the integer multiples of p The range of h spans the set of all nonnegative reals
Figure B-24
Illustration for the solution to Problem 3 in Chap 15 Each horizontal division represents p /2 units Each vertical division represents 1/2 unit The vertical dashed lines are asymptotes of h The positive dependent-variable axis is also an asymptote of h
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Each horizontal division is p /2 units
Each vertical division is 1/2 unit
548 Worked-Out Solutions to Exercises: 11-19
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Each horizontal division is p /2 units Each vertical division is 1 unit
Figure B-25
Illustration for the solution to Problem 4 in Chap 15 Each horizontal division represents p /2 units Each vertical division represents 1 unit The vertical dashed lines are asymptotes of h The dependent-variable axis is also an asymptote of h
4 The dashed gray curves in Fig B-25 are the superimposed graphs of the secant and cosecant functions The complex of solid black curves is a graph of the difference function h (q) = sec q csc q The domain of h includes all real numbers except the integer multiples of p /2 The range of h spans the set of all real numbers 5 The dashed gray curves in Fig B-26 are the superimposed graphs of the secant-squared and cosecant-squared functions The complex of solid black curves is a graph of the difference function h (q) = sec2 q csc2 q The domain of h includes all real numbers except the integer multiples of p /2 The range of h includes all real numbers
15
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Each horizontal division is p /2 units Each vertical division is 1 unit
Figure B-26
Illustration for the solution to Problem 5 in Chap 15 Each horizontal division represents p /2 units Each vertical division represents 1 unit The vertical dashed lines are asymptotes of h The dependent-variable axis is also an asymptote of h
6 We want to find a graph of the ratio function h (q) = (csc2 q)/(sec2 q) We can simplify this function with some algebra, along with our knowledge of trigonometry The secant is the reciprocal of the cosine, so the converse is also true We have 1/(sec q) = cos q Squaring both sides, we get 1/(sec2 q) = cos2 q Substituting in the equation for our original ratio function, we get h (q) = (csc2 q)(cos2 q)
550 Worked-Out Solutions to Exercises: 11-19
The cosecant is the reciprocal of the sine, so csc q = 1/(sin q) Squaring both sides gives us csc2 q = 1/(sin2 q) Substituting in the modified equation for our original function, we obtain f (q) = [1/(sin2 q)] (cos2 q) = [(cos q)/(sin q)]2 The cosine divided by the sine is the cotangent, so we can substitute again to conclude that our original function is h (q) = cot2 q with the restriction that we can t define h for any input value where either the secant or the cosecant become singular The solid black curves in Fig B-27 show the result of squaring all the values of the cotangent function, noting the undefined values as asymptotes or open circles The domain includes all real numbers except integer multiples of p /2, where one or the other of the original squared functions is singular The range is the set of all positive reals
h(q )
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