Polar Coordinates Functions

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  Precalculus: Mathematics for Calculus, International Metric Edition

  • James Stewart, Lothar Redlin, Saleem Watson(Authors)
  • 2016(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  In Section 1.9 we learned how to graph points in rectangular coordinates. In this chapter we study a different way of locating points in the plane, called polar coordinates. Using rectangular coordinates is like describing a location in a city by saying that it’s at the corner of 2nd Street and 4th Avenue; these directions would help a taxi driver find the location. But we may also describe this same location “as the crow flies”; we can say that it’s 1.5 miles northeast of City Hall. These directions would help an airplane or hot air balloon pilot find the location. So instead of specifying the location with respect to a grid of streets and avenues, we specify it by giving its distance and direction from a fixed reference point. That’s what we do in the polar coordinate system. In polar coordinates the location of a point is given by an ordered pair of numbers: the distance of the point from the origin (or pole) and the angle from the positive x-axis. Why do we study different coordinate systems? It’s because certain curves are more naturally described in one coordinate system rather than another. For example, in rectangular coordinates lines and parabolas have simple equations, but equations of circles are rather complicated. We’ll see that in polar coordinates circles have very simple equations. 587 Polar Coordinates and Parametric Equations 8 8.1 Polar Coordinates 8.2 Graphs of Polar Equations 8.3 Polar Form of Complex Numbers; De Moivre’s Theorem 8.4 Plane Curves and Parametric Equations FOCUS ON MODELING The Path of a Projectile © gary718/Shutterstock.com Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience.

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  Sixth Form Pure Mathematics

  Volume 2

  • C. Plumpton, W. A. Tomkys(Authors)
  • 2014(Publication Date)
  • Pergamon

    (Publisher)

  CHAPTER XV POLAR COORDINATES 15.1 Definitions Thus far we have fixed the position of a point in a plane by means of cartesian coordinates only. There are many curves, however, for which cartesian equations are algebraically clumsy and in such cases different coordinate systems are used. One of these systems is provided by Polar Coordinates. In this system the position of a point is fixed by reference to its distance from a fixed point called the Pole, and to the angular displacement of the line which joins it to the pole from a fixed line through the pole, which is called the Initial Line. In Fig. 136, 0 is the pole and 01 the initial line. If OP = r and L 10P = Θ, the polar coordinates of P are (r, 0). FIG. 136. With this definition r is a scalar and any point of the plane has a unique pair of coordinates. In two cases, however, variations from the definitions are conventionally accepted. (i) Negative values of r are defined as values of r measured from the pole in a direction opposite to the direction of r-positive. Thus, the coordinate pairs (r lt 6^, ( — r lt n -f θ χ ) represent the same point. This extension of the definition allows such an equation as r = 1 + 2 cos0 to be represented by a curve which includes the negative values of r arising from the equation. (ii) For curves such as r — kB, values of θ > 2π are considered. A point of the curve, therefore, which, considered merely as a point in 162 PURE MATHEMATICS the coordinate plane, has coordinates (r 1? θ χ ), when referred to the equa-tion, can have coordinates (r l9 θ 1 -f 2ηπ) where n is a positive integer. With this extension of the definition no point of the plane has a unique coordinate pair, but any one of its coordinate pairs defines the point uni-quely. In the light of such variations it is necessary, when working with polar coordinates, to consider many of the problems from first principles.

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  Fundamentals of Modern Mathematics

  A Practical Review

  • David B. MacNeil(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  From the table relating cylindrical and Cartesian coordinates

  From the table z = z

  Thus, substituting these values in the cylindrical equation

  78. The polar coordinate system is essentially a two-dimensional form of either the spherical polar or cylindrical system.Polar coordinates provide an alternate to the Cartesian method of designating points in the XY-plane. It has already been shown how the spherical polar system can be adapted to a two dimensional representation, i.e., ϕ was set equal to 90° and we obtained x = r cos θ; y = r sin θ; z = 0. Similarly, the same system can be derived from the cylindrical coordinate system by setting z = 0. This yields x = ρ cos θ and y = ρ sin θ.

  In this case ρ and r are equivalent.

  Thus we have the following relationships between Cartesian and polar coordinates.

  Example 1.Transform y2 = 8 + 4x into polar coordinates

  From (7-14) , y = ρ sin θ and x = ρ cos θ

  Substituting, ρ2 sin2 θ = 8 + 4ρ cos θ.

  Example 2.Transform into Cartesian coordinates.

  Transposing, ρ(1 − sin θ) = 4

  Substituting from (7-14)

  so that .

  79. Hyperbolic Functions.There is one other system of coordinates to be discussed in this chapter, the bipolar coordinates. Since they are expressed in hyperbolic functions, the latter must first be explained.

  FIG . 7-4. Graphs of e

  x

  and e

  −x

  which define the hyperbolic functions.

  The hyperbolic functions are defined as follows (See Fig. 7-4 ):

  Since Then,

  and the other hyperbolic functions may be computed from these values of sinh and cosh. In using these functions in other computations, their values are found from tables, just as are those of the trigonometric functions.

  In fact, the hyperbolic functions are closely related to the trigonometric functions, having the same relation to the rectangular hyperbola that the trigonometric functions do to the circle. Further relationships are obtained by recalling the expressions for series computation of the trigonometric sin and cosine, which are

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  Fast Start Integral Calculus

  • Daniel Ashlock(Author)
  • 2022(Publication Date)
  • Springer

    (Publisher)

  2 p 2; 2 p 2/. ˙ One of the major uses for converting between the two coordinate systems is to permit us to plot polar points on normal graph paper when we are graphing a polar function. If you have a good eye, or a protractor, it is possible to plot polar points directly, but typically a person just learning polar coordiantes have far more practice plotting .x; y/-points. One of the nice things about polar coordinates is that they let us deal very easily with circles centered at the origin. Circles centered at the origin are constant functions in polar coordinates. The next example demonstrates this. Example 2.25 Graph the polar function r D 2. Solution: r D 2 (0,3) (0,-3) (3,0) (-3,0) ˙ 64 2. PARAMETRIC, POLAR, AND VECTOR FUNCTIONS We usually write polar functions in the form r D f . / making the angle the independent variable and the r the dependent variable. This makes it very easy to give polar functions as parametric functions. Knowledge Box 2.4 Parametric form of polar curves If r D f . / on 1 2 is a polar curve, then a parametric form for the same curve is: .f .t/ cos.t/; f .t/ sin.t// for t 2 OE 1 ; 2 . So far in this section we have established the connections between polar coordinates and the rest of the systems developed in this text. It is time to display polar curves that have unique characteristics that are most easily seen in the polar system. Example 2.26 Graph the polar function r D cos.3 / on OE 0; /. Solution: (0,1) (0,-1) (1,0) (-1,0) The arrows show the drawing direction. ˙ 2.2. POLAR COORDINATES 65 Definition 2.3 Petal curves are curves with equations of the form: r D cos.n / or r D sin.n /; where n is an integer. If no restriction is placed on n, then the curve is traced out an infinite number of times. For odd n, a domain of 2 OE 0; / traces the curve once; when n is even, 2 OE 0; 2 / is needed to trace the entire curve once. Example 2.27 Compare the curves r D sin.5 / and r D cos.5 / on the range 2 OE 0; /.

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  Trigonometry

  • Charles P. McKeague(Author)
  • 2020(Publication Date)
  • XYZ Textbooks

    (Publisher)

  The polar coordinate system also allows us to locate points in the plane but by a slightly different method. Definition (Polar Coordinates) The ordered pair ( r , θ ) names the point that is r units from the origin along the number line (polar axis) that has been rotated through and angle θ from the positive x -axis. The coordinates r and θ are said to be the polar coordinates of the point they name. In polar coordinates, the origin is sometimes referred to as the pole. Graphing in polar coordinates will be a little easier if we revise our coordinate system somewhat. It helps to have angles that are multiples of 15°, along with circles centered at the origin with radii of 1, 2, 3, 4, 5, and 6. A Graphing Polar Coordinates 8.5 Videos Example 1 Graph the points (3, 45°), (2, 120°), ( − 4, 60°), and ( − 5, 150°) on a polar coordinate system. Solution To graph (3, 45°), we locate the point that is 3 units from the origin along the terminal side of 45°. FIGURE 1 y x (3, 45°) 45° 398 Chapter 8 Complex Numbers and Polar Coordinates The point (2, 120°) is 2 units out on the terminal side of 120°. FIGURE 2 y x 120° (2, 120°) As you can see from Figures 1 and 2, if r is positive, we locate the point ( r , θ ) along the terminal side of θ . The next two points we will graph have negative values of r . To graph a point ( r , θ ) in which r is negative, we look for the point on the projection of the terminal side of θ through the origin. To graph ( − 4, 60°), we locate the point that is 4 units from the origin on the projection of 60° through the origin. FIGURE 3 projection of 60° through the origin y x 60° ( 2 4, 60°) To graph ( − 5, 150°), we look for the point that is 5 units from the origin along the projection of 150° through the origin. 8.5 Polar Coordinates 399 y x ( 2 5, 150°) FIGURE 4 In rectangular coordinates, each point in the plane is named by a unique ordered pair ( x , y ). That is, no point can be named by two different ordered pairs.

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  Introduction to Precalculus Analysis, An

  • (Author)
  • 2014(Publication Date)
  • Learning Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter 7 Polar Coordinate System and Mathematical Induction Polar coordinate system Points in the polar coordinate system with pole O and polar axis L . In green, the point with radial coordinate 3 and angular coordinate 60 degrees, or (3,60°). In blue, the point (4,210°). In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction. The fixed point (analogous to the origin of a Cartesian system) is called the pole , and the ray from the pole with the fixed direction is the polar axis . The distance from the pole is called the radial coordinate or radius , and the angle is the angular coordinate , polar angle , or azimuth . ________________________ WORLD TECHNOLOGIES ________________________ History The concepts of angle and radius were already used by ancient peoples of the 1st millennium BCE. The Greek astronomer and astrologer Hipparchus (190-120 BCE) created a table of chord functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions. In On Spirals , Archimedes describes the Archimedean spiral, a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system. From the 8th century CE onward, Muslim astronomers developed methods for appro-ximating and calculating the direction to Makkah (qibla)—and its distance—from any location on the Earth. From the 9th century onward they were using spherical trigo-nometry and map projection methods to determine these quantities accurately.

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  Fundamental Maths

  For Engineering and Science

  • Mark Breach(Author)
  • 2017(Publication Date)
  • Red Globe Press

    (Publisher)

  The Cartesian x -and y -coordinates of a point are measurements from the y -and x -axis respectively. Polar coordinates of a point are the distance of the point from the origin and the orientation of the line joining the origin and the point, with the x -axis, ð r , Þ . There is a direct connection between Cartesian coordinates and polar coordinates, as can be seen from Figure 20.20. The x -coordinate is r cos and the y -coordinate is r sin . Therefore ð x , y Þ¼ð r cos , r sin Þ and so ð r , Þ¼ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 þ y 2 q , tan 1 y x Þ . Note that specifying values for x and y defines only one point, as does specifying r and . Also, a given point only has one set of values for x and y . However the same point can have four different sets of values in polar coordinates. For example if the Cartesian coordinates of a point are ð ffiffiffi 3 p , 1 Þ then the polar coordinates would be ð 2, 30 8 Þ . Coordinate systems 227 x -axis y -axis θ r r cos θ r sin θ ( r , θ ) Figure 20.20 However the polar coordinates could also be expressed as ð 2, 330 8 Þ because 30 8 clockwise from the x -axis brings you to the same orientation as 330 8 anticlockwise from the x -axis. Also, a rotation of 180 8 gives orientation in the opposite direction, as does multiplying the value of r by 1. So, if orientation to the opposite direction is applied twice, that brings us back to where we started. Therefore, since 30 8 þ 180 8 ¼ 210 8 and 330 8 þ 180 8 ¼ 150 8 , then: ð 2, 30 8 Þ is the same point as ð 2, 210 8 Þ and ð 2, 330 8 Þ is the same point as ð 2, 150 8 Þ . Therefore all four sets of polar coordinates describe the same point. Your calculator should have a Polar-to-Rectangular function and vice versa. Different calculators will work differently so read your manual to find how to convert polar coordinates to rectangular (Cartesian) coordinates and vice versa. CAUTION Make sure your calculator is set to the correct angular mode – degrees, radians or grads – before starting your calculations.

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  Engineering Mathematics

  A Programmed Approach, 3th Edition

  • C W. Evans, C. Evans(Authors)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  r ⩾ 0. Of course if we increase θ by 2π, a whole revolution, then we shall obtain the same point as before. In order to establish a unique representation we restrict θ so that 0 ⩽ θ < 2π.

  Fig. 3.2 The polar system.

  There is a minor problem when r = 0, since we then lose our one-to-one correspondence between points in the plane and ordered pairs of real numbers of the form (r, θ). For example (0, π) and (0, π/2) both correspond to the origin. One way of avoiding this problem is to insist that if r = 0 then the origin will be the unique point (0, 0): r = 0, θ = 0. However, we shall not do this as the procedure creates more difficulties than it resolves. Instead we shall avoid representing the origin and insist that r > 0.

  In fact the convention 0 ⩽ θ < 2π is only used occasionally in coordinate geometry. Unfortunately we shall adopt a different convention, namely –π < θ ⩽ π, when we deal with complex numbers (Chapter 10 ). The causes for this are historical and not mathematical, and this goes some way towards explaining why they are illogical.

  3.2      CIRCULAR FUNCTIONS

  It is possible to define the circular functions cos θ and sin θ for any angle θ by using cartesian coordinate geometry and a circle centred at the origin with radius r. In Fig. 3.3 , let X be the point where the circle crosses the positive x-axis, and let the point P on the circle be such that ∠XOP = θ. If P is the point (x, y) then OP = r > 0 and

  cos θ     =

  x r

            sin   θ =  

  y r

  It follows immediately that

  1  If 0 < θ < π/2 then cos θ > 0 and sin θ > 0;

  2  If π/2 < θ < π then cos θ < 0 and sin θ > 0;

  Fig. 3.3

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  Barron's Math 360: A Complete Study Guide to Pre-Calculus with Online Practice

  • Lawrence S. Leff, Christina Pawlowski-Polanish, Barron's Educational Series, Elizabeth Waite(Authors)
  • 2021(Publication Date)
  • Barrons Educational Services

    (Publisher)

  STUDY UNIT IV

  POLAR COORDINATES AND CONIC SECTIONS

  Passage contains an image

  13 POLAR COORDINATES AND PARAMETRIC EQUATIONS

  WHAT YOU WILL LEARN

  Up to now, a curve has been defined by a single equation in two variables and then graphed using rectangular coordinates. In certain situations, however, a curve may not provide a complete picture of the real-world situation it is modeling. For example, when the vertical position of a moving object is plotted against the horizontal position of the object, we can tell the position of the object from the graph, but not the time at which the object was at that position. In such situations, it is helpful to define a curve using two equations by writing x as a function of a third variable, say t, and also writing y as a function of t. The equations x = f(t) and y = g(t) define a curve parametrically in the coordinate plane.

  The polar coordinate system locates a point P in the plane using the ordered pair (r,θ), where the initial side of ∠θ is on the positive x-axis and its vertex, O, is fixed at the origin. The terminal side of ∠θ is the line segment OP, where r = OP, as shown in the accompanying figure.

  Connections between polar coordinates and graphs of complex numbers lead to relationships that allow us to find the nth roots of a complex number.

  LESSONS IN CHAPTER 13

  •Lesson 13-1: Parametric Equations

  •Lesson 13-2: The Polar Coordinate System

  •Lesson 13-3: The Polar Form of a Complex Number

  •Lesson 13-4: Powers and Roots of Complex Numbers

  Lesson 13-1: Parametric Equations

  KEY IDEAS

  A plane curve can be described by a function that uses three variables instead of two. The extra variable, typically represented by t or θ, is called the parameter and provides additional information about the process or function represented by the curve.

  DEFINING A CURVE PARAMETRICALLY

  Suppose that a particle is moving within the coordinate plane in such a way as to trace out the graph of y = x2 − 2x. From this function we know that at, say, x = 5, y = 15. However, we do not know from this function when the particle was at (5,15). Although the function allows us to determine the points where the particle has been, it does not tell us when

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  Student Solutions Manual Analytic Trigonometry with Applications

  • Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Chapter 7 Polar Coordinates; Complex Numbers 327 Chapter 7 Polar Coordinates; Complex Numbers EXERCISE 7.1 Polar and Rectangular Coordinates 1. r gives the distance of P from the pole (origin). 3. Q is the reflection of P across the polar axis (x axis). 5. 7. 9. 11. 13. 15. 17. Exercise 7.1 Polar and Rectangular Coordinates 328 19. Yes. Any point can be represented by (r, ) and by (r,  + 2), among other possibilities. 21. Given P(x, y), then one pair of polar coordinates can be found by taking r = x 2 + y 2 and if x  0 and y  0,  as a solution of tan  = y x . The appropriate solution is determined by the quadrant in which P lies. If x = 0, then if P is on the positive y axis,  =  2 , and if P is on the negative y axis,  = 3 2 . If y = 0, then if P is on the positive x axis,  = 0 and if P is on the negative x axis,  = . 23. Use x = r cos , y = r sin  x = 5 cos  y = 5 sin  = –5 = 0 Rectangular coordinates: (–5, 0) 25. Use x = r cos , y = r sin  x = 3 2 cos –  4       y = 3 2 sin –  4       = 3 2 1 2       = 3 2 – 1 2       = 3 = –3 Rectangular coordinates: (3, –3) 27. Use x = r cos , y = r sin  x = 6 cos 2 3 y = 6 sin 2 3 = 6  1 2       = 6 3 2         = –3 = 3 3 Rectangular coordinates: (–3, 3 3 ) 29. Use x = r cos , y = r sin  x = –3 cos 90° y = –3 sin 90° = 0 = –3 Rectangular coordinates: (0, –3) 31. Use r 2 = x 2 + y 2 and tan  = y x r 2 = 25 2 + 0 2 = 625 r = 25 tan  = 0 25 = 0  = 0 Polar coordinates: (25, 0) 33. Use r 2 = x 2 + y 2 and tan  = y x r 2 = 0 2 + 9 2 = 81 r = 9 tan  = 9 0 = 0 is undefined  =  2 since the point is on the positive y axis Polar coordinates: 9,  2       35. Use r 2 = x 2 + y 2 and tan  = y x r 2 = (– 2 ) 2 + (– 2 ) 2 = 4 r = 2 tan  = – 2 – 2 = 1  = – 3 4 since the point is in the third quadrant Polar coordinates: 2, – 3 4       37.

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