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8 Look back at Fig 15-14, which shows the graphs of the tangent-squared and cotangentsquared functions (dashed gray curves) along with the graph of their sum (solid black curves) Sketch a graph, and state the domain and the range, of the difference-ofsquares function: h (q) = tan2 q cot2 q 9 In Chap 2, we learned that square of the secant of an angle minus the square of the tangent of the same angle is always equal to 1, as long as the angle is not an odd-integer multiple of p /2 That is, sec2 q tan2 q = 1 Sketch a graph that illustrates this principle, which is sometimes called the Pythagorean theorem for the secant and tangent 10 In Chap 2, we learned that the square of the cosecant of an angle minus the square of the cotangent of the same angle is always equal to 1, as long as the angle is not an integer multiple of p That is, csc2 q cot2 q = 1 Sketch a graph that illustrates this principle, which is sometimes called the Pythagorean theorem for the cosecant and cotangent
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CHAPTER
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Parametric Equations in Two-Space
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In the two-space relations and functions we ve seen so far, the value of one variable depends on the value of the other variable In this chapter, we ll learn how to express two-space relations and functions in which both variables depend on an external factor called a parameter
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What s a Parameter
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In a two-space relation or function, a parameter acts as a master controller for one or both variables When there exists a relation between x and y, for example, we don t have to say that x depends on y or vice versa Instead, we can say that a parameter, which we usually call t, independently governs the values of x and y We use parametric equations to describe how this happens
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A rectangular-coordinate example Here s an example of a pair of parametric equations that produce a straight line in the Cartesian xy plane Consider
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x = 2t and y = 3t To generate the graph of this system, we can input various values of t to both of the parametric equations, and then plot the ordered pairs (x,y) that come out Following are some examples:
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When t = 2, we have x = 2 ( 2) = 4 and y = 3 ( 2) = 6 When t = 1, we have x = 2 ( 1) = 2 and y = 3 ( 1) = 3 When t = 0, we have x = 2 0 = 0 and y = 3 0 = 0 When t = 1, we have x = 2 1 = 2 and y = 3 1 = 3 When t = 2, we have x = 2 2 = 4 and y = 3 2 = 6
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What s a Parameter
When we plot the (x,y) ordered pairs based on the above list as points on a Cartesian plane and then connect the dots, we get a line passing through the origin with a slope of 3/2, as shown in Fig 16-1 From our knowledge of the slope-intercept form of a line in the xy plane, we can write down the equation in that form as y = (3/2)x Alternatively, we can use algebra to derive the equation of our system in terms of x and y alone, without t Let s take the first parametric equation x = 2t and multiply it through by 3/2 to get (3/2)x = (3/2)(2t) = 3t Deleting the middle portion in the above three-way equation gives us (3/2)x = 3t The second parametric equation tells us that 3t = y, so we can substitute directly in the above equation to obtain (3/2)x = y
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