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We can simplify this problem by remembering a few basic facts in trigonometry, and by applying a little algebra First, let s remember that the cosecant is equal to the reciprocal of the sine, so the converse is also true We have 1/(csc q) = sin q When we square both sides, we get 1/(csc2 q) = sin2 q Substituting in the equation for our function gives us f (q) = (sec2 q) (sin2 q) We ve learned that the secant is equal to the reciprocal of the cosine We have sec q = 1/(cos q) so we can square both sides to get sec2 q = 1/(cos2 q)
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Graphs Involving the Secant and Cosecant
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Substituting again in the equation for our original function, we obtain f (q) = (sin2 q)/(cos2 q) = [(sin q)/(cos q)]2 The sine over the cosine is equal to the tangent, so we can substitute again to conclude that our original function is f (q) = tan2 q with the restriction that we can t define it for any input value where either the secant or the cosecant become singular The solid black curves in Fig 15-10 show the result of squaring all the values of the tangent function, noting the additional undefined values as open circles At the points shown by the open circles, the cosecant function is singular so its square is undefined That means we can t define our ratio function f at any such point At the asymptotes (dashed vertical lines), the secant function is singular so its square is undefined, making it impossible to define the ratio function f for those values of q Our function f has a period of p The domain of f 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
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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 unit
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Figure 15-10 Graph of the ratio of the square of the
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secant function to the square of the cosecant function (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
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Graphs Involving the Tangent and Cotangent
You were introduced to graphs of the basic tangent and cotangent functions in Chap 2 The tangent is the ratio of the sine to the cosine, and the cotangent is the ratio of the cosine to the sine Now we ll see what happens when we alter or combine these functions in a few different ways
Tangent and cotangent: example 1 In Fig 15-11, the dashed gray curves are superimposed graphs of the tangent and cotangent functions The solid black curves portray the graph of
f (q) = tan q + cot q The graph of f has asymptotes that pass through every point where the independent variable attains an integer multiple of p /2 The period is p The domain is the set of all real numbers except the integer multiples of p /2 The range is the set of reals larger than or equal
f (q )
Each horizontal division is p /2 units Each vertical division is 1 unit
Figure 15-11
Graph of the sum of the tangent and cotangent 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 dependent-variable axis is also an asymptote of f
Graphs Involving the Tangent and Cotangent
to 2 or smaller than or equal to 2 We can also say that the range spans the set of all reals except those in the open interval ( 2,2)
Tangent and cotangent: example 2 Figure 15-12 shows superimposed graphs of the tangent and cotangent functions (dashed gray curves) along with their product (black line with holes in it) We have
f (q) = tan q cot q We can simplify the calculations to graph this function when we recall that the cotangent and the tangent are reciprocals of each other, so we have cot q = 1/(tan q) This equation is valid as long as both functions are defined and tan q 0 By substitution, the equation for our function f becomes f (q) = (tan q)/(tan q) = 1
f (q )