10Polar Coordinates,Parametric EquationsCoordinate systems are tools that let us use algebraic methods to understand geometry.While therectangular(also calledCartesian) coordinates that we have been using arethe most common, some problems are easier to analyze in alternate coordinate systems.A coordinate system is a scheme that allows us to identify any point in the plane orin three-dimensional space by a set of numbers. In rectangular coordinates these numbersare interpreted, roughly speaking, as the lengths of the sides of a rectangle.Inpolarcoordinatesa point in the plane is identified by a pair of numbers (r, θ). The numberθmeasures the angle between the positivex-axis and a ray that goes through the point,as shown in figure 10.1; the numberrmeasures the distance from the origin to the point.Figure 10.1 shows the point with rectangular coordinates (1,√3) and polar coordinates(2, π/3), 2 units from the origin andπ/3 radians from the positivex-axis.√31..(2, π/3)•.Figure 10.1Polar coordinates of the point (1,√3).217218Chapter 10 Polar Coordinates, Parametric EquationsJust as we describe curves in the plane using equations involvingxandy, so can wedescribe curves using equations involvingrandθ. Most common are equations of the formr=f(θ).EXAMPLE 10.1Graph the curve given byr= 2. All points withr= 2 are at distance2 from the origin, sor= 2 describes the circle of radius 2 with center at the origin.EXAMPLE 10.2Graph the curve given byr= 1 + cosθ.We first considery=1 + cosx, as in figure 10.2. Asθgoes through the values in [0,2π], the value ofrtracksthe value ofy, forming the “cardioid” shape of figure 10.2. For example, whenθ=π/2,r= 1 + cos(π/2) = 1, so we graph the point at distance 1 from the origin along thepositivey-axis, which is at an angle ofπ/2 from the positivex-axis.Whenθ= 7π/4,r= 1 + cos(7π/4) = 1 +√2/2≈1.71, and the corresponding point appears in the fourthquadrant.This illustrates one of the potential benefits of using polar coordinates:theequation for this curve in rectangular coordinates would be quite complicated.012π/2π3π/22π••.••..........Figure 10.2A cardioid:y= 1 + cosxon the left,r= 1 + cosθon the right.Each point in the plane is associated with exactly one pair of numbers in the rect-angular coordinate system; each point is associated with an infinite number of pairs inpolar coordinates. In the cardioid example, we considered only the range 0≤θ≤2π, andalready there was a duplicate: (2,0) and (2,2π) are the same point. Indeed, every value ofθoutside the interval [0,2π) duplicates a point on the curver= 1+cosθwhen 0≤θ <2π.We can even make sense of polar coordintes like (-2, π/4): go to the directionπ/4 andthen move a distance 2 in the opposite direction; see figure 10.3. As usual, a negative angleθmeans an angle measured clockwise from the positivex-axis. The point in figure 10.3also has coordinates (2,5π/4) and (2,-3π/4).