Polar Coordinates

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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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  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)

  Blaise Pascal subsequently used Polar Coordinates to calculate the length of parabolic arcs. In Method of Fluxions (written 1671, published 1736), Sir Isaac Newton examined the transformations between Polar Coordinates, which he referred to as the Seventh Manner; For Spirals, and nine other coordinate systems. In the journal Acta Eruditorum (1691), Jacob Bernoulli used a system with a point on a line, called the pole and polar axis respectively. Coordinates were specified by the distance from the pole and the angle from the polar axis . Bernoulli's work extended to finding the radius of curvature of curves expressed in these coordinates. The actual term Polar Coordinates has been attributed to Gregorio Fontana and was used by 18th-century Italian writers. The term appeared in English in George Peacock's 1816 translation of Lacroix's Differential and Integral Calculus . Alexis Clairaut was the first to think of Polar Coordinates in three dimensions, and Leonhard Euler was the first to actually develop them. ________________________ WORLD TECHNOLOGIES ________________________ Common conventions A polar grid with several angles labeled in degrees The radial coordinate is often denoted by r , and the angular coordinate by θ or t . Angles in polar notation are generally expressed in either degrees or radians (2π rad being equal to 360°). Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics and mathematical physics. In many contexts, a positive angular coordinate means that the angle θ is measured counterclockwise from the axis. In mathematical literature, the polar axis is often drawn horizontal and pointing to the right. Uniqueness of Polar Coordinates Adding any number of full turns (360°) to the angular coordinate does not change the corresponding direction. Also, a negative radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction.

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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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  Foundations of Mathematics

  Algebra, Geometry, Trigonometry and Calculus

  • Philip Brown(Author)
  • 2016(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  section 2.6 ).

  The mathematical methods in a two-dimensional coordinate system that we introduce in this chapter can be generalized to a three-dimensional coordinate system (although we will not do it in this book). A three-dimensional coordinate system can be used as a mathematical representation of the three-dimensional physical space in which we humans move around; and it is in this coordinate representation of three-dimensional space that mathematicians and scientists can carry out advanced mathematical simulations of dynamic processes like ocean currents, planetary weather patterns, motions of projectiles, and exploding stars.

  2.2WORKING IN A COORDINATE SYSTEM

   

  Any given point in the Cartesian plane can be labeled by means of two numbers called coordinates. The first coordinate is called the x-coordinate, and the second is called the y-coordinate. The x- and y-coordinates together form a coordinate pair. The x-coordinate can be found by following a vertical line from the given point to a point on the x-axis and reading its position on the number line. Similarly, the y-coordinate can be found by following a horizontal line from the given point to a point on the y-axis and reading its position on the number line. Points in the Cartesian plane are usually labeled using uppercase letters with the coordinate values following in parentheses, as shown in figure 2.2 . The axes separate the Cartesian plane into four quadrants (first, second, third, and fourth quadrants) that can be labeled as I, II, III, and IV, respectively, in a counterclockwise order, starting with the upper right quadrant.

  FIGURE 2.2. Points in the Cartesian plane.

  2.3LINEAR EQUATIONS AND STRAIGHT LINES

   

  Much of mathematics involves the study of related quantities in the form of equations. For this reason, we begin with a study of linear equations and the graphs (infinite straight lines) of linear equations in the Cartesian plane because these quantify the simplest kind of relationship (linear) that related quantities can have. We will refer to an “infinite straight line” (in the Cartesian plane) as a “line.”

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  Mathematics for Scientific and Technical Students

  • H. Davies, H.G. Davies, G.A. Hicks(Authors)
  • 2014(Publication Date)
  • Routledge

    (Publisher)

  coordinates, are required to locate a point in a plane. Two systems are used:

  (a) Cartesian coordinates (x , y )

  This is the most commonly used system. Two perpendicular datum lines are used, the horizontal line is called the x -axis, the vertical line is called the y -axis, as shown in Fig. 9.1 . The point of intersection of the two axes is called the origin O. Any point P is located by its perpendicular distance from the two axes.

  Fig. 9.1

  (b) Polar Coordinates (r , θ )

  In this system a point P is located at a distance r along a line OP from a fixed point O , called the pole, as shown in Fig 9.2 . θ is the angle that the line OP makes with the reference + x -axis. It is important to remember that θ is positive when OP rotates anticlockwise.

  Fig. 9.2

  Fig. 9.3

  Fig. 9.3 shows the relationship between both systems. Pythagoras’s theorem and trigonometry can be used to change from one system to another.

  (c) Conversion from Cartesian to Polar Coordinates

  From Pythagoras

  and

  The smallest value of θ is usually quoted and can be positive or negative. The value of θ obtained must be checked so that it places P in the correct quadrant. This can be done by using a sketch to check the results, as shown in Example 9.1 .

  (d) Conversion from polar to Cartesian coordinates

  From Fig. 9.3 , using trigonometrical ratios in a right-angled triangle:

  Example 9.1 The Cartesian coordinates of a point P are (–4, –6). Convert these to Polar Coordinates.

  The point P with these two coordinates is shown in Fig. 9.4 .

  Fig. 9.4

  The value of 56.3° is not in agreement with the position of P in the third quadrant. The smallest magnitude of θ is 180 – 56.3 = 123.7°, and this is seen to be negative because it is in a clockwise direction.

  The Polar Coordinates are (7.2, –124°).

  Example 9.2 Convert the Polar Coordinates (18, 125°) to Cartesian coordinates.

  Fig. 9.5

  From Fig. 9.5 ,

  The Cartesian coordinates are (–10.3, 14.7).

  9.3 The distance between two points

  Fig. 9.6 shows two points P (x 1 , y 1 ) and Q (x 2 , y 2 )

  Horizontal distance

  Fig. 9.6

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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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  Classical Mechanics

  • A. Douglas Davis(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  6.3 Cylindrica l Pola r Coordinate s We ca n easil y exten d ou r notatio n t o thre e dimension s b y addin g a z-componen t an d creatin g cylindrical pola r coordinate s a s sketche d in Figur e 6.3.1 . æ Figur e 6.3. 1 Cylindrica l pola r coordinates . CHAPTE R 6 / COORDINAT E SYSTEM S 6.4 Spherica l Pola r Coordinate s On e o f th e mos t usefu l coordinat e system s is spherical Polar Coordinates (ofte n simpl y calle d spherical coordinates), a s show n in Figur e 6.4.1 . Suc h a syste m is usefu l wheneve r ther e is spherica l symmetry—fo r example , describin g th e force s bindin g a n electro n t o it s nucleus , givin g th e locatio n o f th e spac e shuttl e in orbi t aroun d Earth , o r chartin g th e cours e o f th e U.S.S. Enterprise throug h th e galaxy . In Figur e 6.4.1 , poin t Ñ is locate d b y givin g it s radial distance fro m th e origi n r, an d azimuthal angle ö locatin g a plan e whos e angl e of rotatio n is measure d fro m th e x-axis , an d a polar angle 0 , givin g it s angula r locatio n measure d down fro m th e z-axis . Not e tha t 0 ca n var y fro m 0 t o ð , an d ö fro m 0 t o 2ð . Thi s is sketche d in Figur e 6.4.1 . We ca n the n defin e mutuall y perpendicula r uni t vector s f, È, an d a s show n in Figur e 6.4. 1 b y givin g th e direction s tha t Ñ move s a s r, 0, an d ö increase , respectively . Thes e uni t vector s ar e sketche d agai n in Figur e 6.4.2 . Not e tha t f an d è bot h lie in a plan e containin g th e z-axi s an d rotate d a n amoun t ö fro m th e x-axis . Th e uni t vecto r cji lie s in th e xy-plane . Figur e 6.4. 3 Th e locatio n o f a poin t Ñ is give n b y it s radia l distanc e fro m th e z-axis , p , its angula r rotatio n fro m th e x-axis , ö (jus t a s in th e previou s cas e wit h r an d 0), and it s elevatio n abov e th e xy plane , z.

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