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1 Figure A-3 shows the graphs of the equations q = p /4 and q = p /2 in polar coordinates, where q is the independent variable and r is the dependent variable Neither of these are functions of q In the first case, r can be any real number when q = p /4 In the second case, r can be any real number when q = p /2 2 The graph of q = p /4 is a sloping line through the origin in the Cartesian xy plane The graph of q = p /2 is a vertical line that coincides with the y axis Figure A-4 shows both graphs The line representing q = p /4 portrays a function of x in the Cartesian xy plane, because there is never more than one value of y for any value of x But the line representing q = p /2 does not portray a function of x in the Cartesian xy plane, because when x = 0, y can be any real number Figure A-3
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y 6 4 2
q = p /2
q = p /4
2 2 4 6
476 Worked-Out Solutions to Exercises: 1-9
3 The equation r = a represents the same circle as the equation r = a 4 Imagine a ray that points straight to the right along the reference axis labeled 0 As the ray rotates counterclockwise so that q starts out at 0 and increases positively, the corresponding radius r starts out at 0 and increases negatively This tells us that the constant a is negative When the ray has turned through 1/2 rotation so that q = p, the radius of the solid spiral reaches the value r = 2p (Don t get this confused with the apparent radius of r = 4p on the solid spiral! The larger value is actually r = 4p, which we get when the ray has rotated through a complete circle so that q = 2p) We can solve for a by substituting the number pair (q,r) = (p, 2p) in the general spiral equation r = aq This gives us 2p = ap which solves to a = 2 Therefore, the equation of the pair of spirals is r = 2q 5 Line L runs through the origin and up to the left at an angle halfway between the p /2 axis and the p axis That direction is represented by q = 3p /4 This is the equation of L But we can also imagine that line L runs down and to the right at an angle corresponding to q = 7p /4 so this can also serve as the equation of L Theoretically, we can add or subtract any integer multiple of p from 3p /4 and get a valid equation for L By convention, we stick to the range of angles 0 q < 2p, so the above two equations are preferred over any others Circle C is centered at the origin and has a radius of 3 units, as we can see by inspecting the graph and remembering that each radial division equals 1 unit Therefore, C can be represented by r=3 We can also consider the radius to be 3 units, so r = 3 is an equally valid equation for C
3
6 Based on the solution to Problem 5, we can represent the intersection point at the upper left as either P = (3p /4,3) or P = (7p /4, 3) We can represent the intersection point at the lower right as either Q = (7p /4,3) or Q = (3p /4, 3) The more intuitive representations are the coordinates with positive radii, which are P = (3p /4,3) and Q = (7p /4,3) 7 Before we can solve the system of equations for L and C as they are shown in Fig 3-8, we must be certain that we ve completely identified the system For L, we have q = 3p /4 or q = 7p /4 and for C, we have r=3 or r = 3 Solving this system is deceptively simple It doesn t require algebra at all! We merely combine all the possible combinations of angles and radii we ve listed above to get the following four ordered pairs: (q,r) = (3p /4,3) (q,r) = (3p /4, 3) (q,r) = (7p /4,3) (q,r) = (7p /4, 3)
478 Worked-Out Solutions to Exercises: 1-9
Using plus-and-minus notation for the radii, we can reduce this list to two items: (q,r) = (3p /4, 3) and (q,r) = (7p /4, 3) That s redundant, but it s valid If we want to be more elegant, we can get rid of the redundancy and list the solutions as (q,r) = (3p /4,3) and (q,r) = (7p /4,3) We can tell which ordered pair represents P and which one represents Q by looking again at Fig 3-8 It s obvious that P = (3p /4,3) and Q = (7p /4,3) 8 Let s take away the polar grid in Fig 3-8 and put a Cartesian grid in its place, as shown in Fig A-5 Because we ve been told that line L is equally distant from the vertical and