Polar Curves
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Complete description on polar curves Relation between polar and rectangular coordinates Polar curves sketching
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This document discusses procedures for tracing polar curves. It explains how to determine symmetry properties, the region over which the curve is defined, tabulation of points, finding the angle between the radius vector and tangent, and identifying asymptotes. Two examples are provided to demonstrate the tracing process. The first example traces the curve r=a(1+cosθ) and finds it is symmetric about the initial line, lies within a circle, and passes through the origin. The second example traces r2=a2cos2θ, finding its symmetries and that the curve exists between 0<θ<π/4 and 3π/4<θ<π.
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Okay, here are the steps to solve this problem:
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This document provides an overview of polar curves and coordinates. It defines polar coordinates, explains how to plot points in a polar coordinate system, and how to convert between polar and rectangular coordinates. Examples are given to plot points with polar coordinates and express rectangular coordinates in terms of polar coordinates. Finally, common polar curves such as the cardiod, limacon, roses, spirals, and lemniscate are listed.
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Okay, here are the steps to solve this problem:
1) The sphere has a radius of 8 inches
2) To find the volume of a sphere we use the formula: V = (4/3)πr^3
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V = (4/3)π(8)^3
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The document describes polar coordinates. Polar coordinates represent the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the positive x-axis and a line from O to P. θ is positive for counter-clockwise angles and negative for clockwise angles. The polar coordinate (r, θ) uniquely identifies the location of P. Polar coordinates (r, θ) can be converted to rectangular coordinates (x, y) using the relations x=rcos(θ) and y=rsin(θ).
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In spherical coordinates, each point is represented by an ordered triple of a distance and two angles, similar to the latitude-longitude system used on Earth. A point P is specified by its coordinates P(r,θ,φ), where r is the distance from the origin and θ and φ are the angular coordinates. Orthogonal surfaces in the spherical coordinate system are generated by keeping r, θ, or φ constant, resulting in a sphere, circular cone, or semi-infinite plane, respectively.
34 polar coordinate and equations
Polar coordinates provide an alternative way to specify the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the x-axis and a line from O to P. θ is measured counter-clockwise from the x-axis and can be either positive or negative. A point P's polar coordinates (r, θ) uniquely identify its location. Polar and rectangular coordinates can be converted between each using the relationships x=r*cos(θ), y=r*sin(θ), and r=√(x2+y2).
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This document provides information about coordinate grids, ordered pairs, and formulas in coordinate geometry. It defines key terms like coordinates, quadrants, and distance and section formulas. The distance formula calculates the distance between two points with coordinates (x1, y1) and (x2, y2). The section formula finds the coordinates of a point that divides a line segment between (x1, y1) and (x2, y2) in a given ratio. It also discusses finding the midpoint and calculating the area of a triangle using coordinates.
Spherical Polar Coordinate System- physics II
- The document introduces polar coordinates as an alternative coordinate system to Cartesian coordinates.
- In polar coordinates, a point P is represented by an ordered pair (r, θ) where r is the distance from a fixed point called the pole (or origin) to P, and θ is the angle between the polar axis and the line from the pole to P.
- Formulas are provided to convert between Cartesian (x, y) coordinates and polar (r, θ) coordinates. Specifically, r2 = x2 + y2 and θ = tan-1(y/x).
- An example shows calculating the polar coordinates (5, 0.93) for the Cartesian point (3, 4).
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- 1. Sarvajanik College Of Engineering And Technology Calculus(2110014) Computer Engineering Eve. (COE) Date Of Submission :- 13/11/2017
- 2. Group Members And Topic 55 – Ms. Shakshi Mehta 56 – Mr. Priyank Bardoliya 57 – Mr. Keyur Vadodariya 58 – Ms. Rishika Jain 59 – Mr. Pathik Thakor Topic:- Polar Curves Guided By :- Prof. Dixa P. Kevadiya
- 3. The Polar Co-ordinate System The Polar Coordinates System is a coordinate system in which the coordinates of a point in a plane are its distances from a fixed point and its direction from a fixed line. The fixed point is called the origin or the pole and the fixed line is called the polar axis. The coordinates given in this way are called Polar Coordinates. Where: O - Origin or Pole OA - Polar Axis r - Radius Vector θ - Vectorial Angle P(r, θ) – Polar Coordinate
- 4. Sign Convection For Vectorial Angle (θ): • Positive (+) – If it is measured counterclockwise from the polar axis. • Negative (-) – If it is measured clock wise from the polar axis. For Radius Vector (r): • Positive (+) – If it is measures from the pole along the terminal side of Vectorial angle (θ). • Negative (-) – If it is measured along the terminal side extended through the pole.
- 5. Examples :- Plot The Given Points On Given Polar Coordinate System 1.P (3, 45°) 2.P (-3, –75°)
- 6. Relation Between Rectangular And Polar Coordinates The transformation formulas that express the relationship between rectangular coordinates and polar coordinates of a point are as follows:
- 7. Example :- Find the rectangular coordinates of the points defined by the polar coordinates (6,150°).
- 8. Graph Of Polar Equation Eight-Leaf Rose The graph of an equation r = f(θ) in polar coordinates is the set of all points (r, θ) whose coordinates satisfy the equation. Special Types Of Polar Curves :- Three-Leaf Rose 1.Rose Curve r = a sin(nθ) or r = a cos(nθ), if n is odd, There are n leaves; if n is even there are 2n leaves.
- 9. 2.Spirals Spiral Of Archimedes; r = aθ Logarithmic Spiral; r = eaθ 3.Limacon r = b + asin(θ) or r = b + acos(θ), if a = b, the Limacon is called a Cardiod.
- 10. Example :- Trace The Curve r = 1+ 2cosθ Table :- Comparing with equation r = b + a cosθ we get b=1 and a=2 i.e. |b| < |a| , so the graph has inner loop while if it was having |b| > |a| , the graph will be the curve surrounding the origin. θ 0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330° r 3 2.73 2 1 0 -0.73 -1 -0.73 0 1 2 2.73