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Each horizontal division is p /2 units Each vertical division is 1 unit
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292 Trigonometric Curves
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Secant and cosecant: example 3 The dashed gray curves in Fig 15-8 are the graphs of the secant function (at A) and the cosecant function (at B) At A, the solid black curve is the graph of
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f (q) = sec2 q At B, the solid black curve is the graph of g (q) = csc2 q
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Figure 15-8 The solid black
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curves are the graphs of the squares of the secant function (at A) and the cosecant function (at B) The dashed gray curves are the 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 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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The squared functions have periods of p, which are half the periods of the original functions Therefore, the frequencies of the squared functions are double those of the originals Singularities occur at the same points on the independent-variable axes as they do for the original functions The domain of the secant-squared function is the set of all reals except odd-integer multiples of p/2 The domain of the cosecant-squared function is the set of all reals except integer multiples of p The ranges in both cases are confined to the set of reals y such that y 1
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Secant and cosecant: example 4 Figure 15-9 shows what happens when we add the secant-squared function to the cosecantsquared function The solid black curves compose the graph of
f (q) = sec2 q + csc2 q The dashed gray curves are superimposed graphs of the original functions This sum function 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 4
Figure 15-9
Graph of the sum of the squares of the secant and cosecant functions (solid black curves) The dashed gray curves are the graphs of the original squared functions Each horizontal division represents p/2 units Each vertical division represents 1 unit The vertical dashed lines are asymptotes of f The positive dependent-variable axis is also an asymptote of f
Each horizontal division is p /2 units
f (q )
Each vertical division is 1 unit
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Are you confused
You re bound to wonder, How do we know that the range of the sum-of-squares function in the previous example is the set of all reals greater than or equal to 4 Another way of stating this fact is that the minima of the solid black curves in Fig 15-9 have dependent-variable values equal to 4 These minima occur at values of q where the graphs of the secant-squared and cosecant-squared functions (dashed gray curves) intersect Every one of those points occurs where q is an odd-integer multiple of p /4 With the help of your calculator, you can determine that whenever q is an odd-integer multiple of p/4, the secant squared and cosecant squared are both 2, so their sum is 4 If you move slightly to the right or left of any of these points, the value of the sum-of-squares function increases (a fact that you can, again, check out with your calculator) It follows that the sum-ofsquares function can never attain any real-number value less than 4 However, there s no limit to how large the value of the function can get One or the other of the original functions blows up positively at every point where q attains an integer multiple of p /2
Here s a challenge!
Sketch a graph of the ratio of the square of the secant function to the square of the cosecant function That is, graph f (q) = (sec2 q)/(csc2 q) Determine the domain and range of f Be careful! Both the domain and the range have some tricky restrictions
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