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.