10 PARAMETRIC EQUATIONS AND POLAR COORDINATES
PARAMETRIC EQUATIONS & POLAR COORDINATES A coordinate system represents apoint in the plane by an ordered pairof numbers calledcoordinates.
PARAMETRIC EQUATIONS & POLAR COORDINATES Usually, we use Cartesian coordinates, which are directed distances from twoperpendicularaxes.
PARAMETRIC EQUATIONS & POLAR COORDINATES Here, we describe a coordinate systemintroduced by Newton, calledthe polarcoordinate system. • It is more convenientfor many purposes.
PARAMETRIC EQUATIONS & POLAR COORDINATES 10.3 Polar Coordinates In this section, we will learn: How to represent points in polar coordinates.
POLE We choose a point in the plane that is called the pole (or origin) and is labeled O.
POLAR AXIS Then, we draw a ray (half-line) starting atO called the polar axis. • This axis is usually drawn horizontallyto theright corresponding to thepositive x-axisinCartesiancoordinates.
ANOTHER POINT If Pis any other point in the plane, let: • rbe thedistance from O to P. • θbe theangle(usually measured in radians) between thepolar axis and the lineOP.
POLAR COORDINATES P is represented by theordered pair (r,θ). r,θ are called polar coordinatesof P.
POLAR COORDINATES We use the convention that an angle is: • Positive—if measured in the counterclockwisedirection from the polar axis. • Negative—if measured inthe clockwise direction from the polar axis.
POLAR COORDINATES If P =O, then r = 0, and we agree that (0, θ) represents the pole for any value of θ.
POLAR COORDINATES We extend the meaning of polar coordinates (r,θ) to the case in which ris negative—as follows.
POLAR COORDINATES We agree that, as shown, the points (–r,θ)and (r,θ) lie on thesame linethroughO and at the samedistance | r | fromO, but on oppositesides of O.
POLAR COORDINATES If r > 0, the point (r, θ) lies in the same quadrant as θ. Ifr < 0, it lies in the quadrant on theopposite side of thepole. • Notice that (–r, θ) represents the same point as (r, θ + π).
POLAR COORDINATES Example 1 Plot the points whose polar coordinates aregiven. • (1, 5π/4) • (2, 3π) • (2, –2π/3) • (–3, 3π/4)
POLAR COORDINATES Example 1 a The point (1, 5π/4) is plotted here.
POLAR COORDINATES Example 1 b The point (2, 3π) is plotted.
POLAR COORDINATES Example 1 c The point (2, –2π/3) is plotted.
POLAR COORDINATES Example 1 d The point (–3, 3π/4) is plotted. • It is is locatedthree units fromthepole in the fourth quadrant. • This is because theangle 3π/4 isinthe secondquadrant and r = -3 isnegative.
CARTESIAN VS. POLAR COORDINATES In the Cartesian coordinate system, everypointhas only one representation. However, inthepolarcoordinate system, each point has manyrepresentations.
CARTESIAN VS. POLAR COORDINATES For instance, the point (1, 5π/4) in Example 1a could be written as: • (1, –3π/4), (1, 13π/4), or (–1, π/4).
CARTESIAN & POLAR COORDINATES In fact, as a complete counterclockwiserotation is given by an angle 2π,the pointrepresented by polar coordinates (r, θ) is alsorepresented by (r, θ + 2nπ) and(-r, θ + (2n + 1)π)where nis any integer.
CARTESIAN & POLAR COORDINATES The connection between polar and Cartesiancoordinates can be seen here. • The pole corresponds to the origin. • The polar axis coincides with the positivex-axis.
CARTESIAN & POLAR COORDINATES If the point P has Cartesian coordinates (x, y) and polar coordinates (r, θ), then,from thefigure, we have:
CARTESIAN & POLAR COORDS. Equations 1 Therefore,
CARTESIAN & POLAR COORDS. Although Equations 1 were deduced from the figure (which illustrates the casewherer > 0 and 0 < θ< π/2), these equations are valid for allvalues of rand θ. • See the generaldefinition of sin θand cos θin Appendix D.
CARTESIAN & POLAR COORDS. Equations 1 allow us to find the Cartesiancoordinates of a point when the polarcoordinatesare known.
CARTESIAN & POLAR COORDS. Equations 2 To find randθ when x andyare known,we use the equations • These can be deduced from Equations 1 or simply read from the figure.
CARTESIAN & POLAR COORDS. Example 2 Convert the point (2, π/3) from polar to Cartesian coordinates. • Sincer = 2 and θ= π/3, Equations 1 give: • Thus, the point is (1, ) in Cartesian coordinates.
CARTESIAN & POLAR COORDS. Example 3 Represent the point with Cartesiancoordinates (1, –1) in terms of polarcoordinates.
CARTESIAN & POLAR COORDS. Example 3 If we choose rto be positive, then Equations 2 give: • As the point (1, –1) lies in the fourthquadrant, we can chooseθ = –π/4or θ = 7π/4.
CARTESIAN & POLAR COORDS. Example 3 Thus,one possible answer is: ( , –π/4) Another possible answer is: ( ,7π/4)
CARTESIAN & POLAR COORDS. Note Equations 2 do not uniquely determine θwhenx and y are given. • This is because, as θincreases through the interval 0 ≤θ≤2π, each value of tan θoccurs twice.
CARTESIAN & POLAR COORDS. Note So, inconverting from Cartesian topolar coordinates, it’s not good enough just tofindr andθthat satisfy Equations 2. • As inExample 3, we must choose θ so that thepoint (r, θ) lies inthe correct quadrant.
POLAR CURVES The graph of a polar equation r = f(θ) [or, more generally, F(r, θ) = 0] consists of allpoints that have atleast one polarrepresentation (r, θ), whosecoordinatessatisfy theequation.
POLAR CURVES Example 4 What curve is represented by the polarequation r = 2? • The curve consists of all points (r, θ) with r = 2. • r represents the distancefrom the point to the pole.
POLAR CURVES Example 4 • Thus, thecurve r = 2 represents the circle with center O andradius 2.
POLAR CURVES Example 4 In general, the equation r = arepresents acircle O with center and radius |a|.
POLAR CURVES Example 5 Sketch the polar curve θ= 1. • This curve consists of all points (r, θ) such that the polar angle θ is 1 radian.
POLAR CURVES Example 5 Itis the straight line that passes through O andmakesan angle of 1 radian with thepolaraxis.
POLAR CURVES Example 5 Notice that: • The points (r, 1) on the line withr > 0 are in the firstquadrant. • The points (r, 1) on the line with r < 0 are in the third quadrant.
POLAR CURVES Example 6 • Sketch the curve with polar equationr = 2 cos θ. • Find a Cartesian equation for this curve.
POLAR CURVES Example 6 a First,we find the values of rfor someconvenient values of θ.
POLAR CURVES Example 6 a We plot thecorresponding points (r, θ). Then, we join thesepoints to sketch the curve—as follows.
POLAR CURVES Example 6 a The curve appears to be a circle.
POLAR CURVES Example 6 a We have used only values of θbetween 0 andπ—since, if we let θ increasebeyond π, weobtain the same points again.
POLAR CURVES Example 6 b To convert the given equation to a Cartesian equation, we use Equations 1 and 2. • Fromx = r cos θ, we have cos θ= x/r. • So, the equation r = 2 cos θ becomes r= 2x/r. • Thisgives: 2x =r2 = x2 + y2 or x2 + y2 – 2x = 0
POLAR CURVES Example 6 b Completing the square, we obtain: (x – 1)2+y2= 1 • The equation is of a circle with center (1, 0) and radius 1.
POLAR CURVES The figure shows a geometrical illustrationthat the circle in Example 6 has the equationr = 2 cosθ. • The angle OPQis a right angle, and so r/2 = cos θ. • Why is OPQa right angle?
POLAR CURVES Example 7 Sketch the curve r = 1 + sinθ. • Here, we do not plot points as in Example 6. • Rather, we first sketch the graph ofr = 1 + sinθin Cartesiancoordinates byshifting the sinecurveup oneunit—as follows.