8.2
Graphs of Polar Equations
A grid with concentric circles and straight radial lines, all centered at a single point called the pole, defines a polar coordinate system.
Each point on this grid is identified by its distance from the pole and by the angle it makes with the polar axis.
A graph where every point stays the same distance from the origin forms a perfect circle. The radius remains constant throughout the rotation.
A straight line in polar coordinates is traced when the angle remains fixed and the radius increases.
When the radius is represented by a cosine function, the path traced by this radius forms a circle with the origin on its edge.
Changing the sign of the constant a flips the circle across the vertical axis.
The polar graphs help in tracing satellite orbits around Earth. Here, r represents the satellite's distance from the centre of the Earth, and θ is the angle measured from a reference line. This allows accurate tracking of the satellite, as its angle values change continuously, providing precise information about its position relative to Earth's surface.
The polar coordinate system represents points using a distance from a central point (the pole) and an angle from a reference direction (the polar axis). Unlike rectangular coordinates, polar coordinates are ideal for graphing curves with radial symmetry or periodic behavior.
Some general forms of graphs in polar coordinates include the following:
Equation of a Circle (Centered at the Pole):
A graph where the radius remains constant for all angles traces a circle centered at the pole:
Equation of a Line Through the Pole:
Holding the angle constant while letting the radius vary forms a straight line:
Equation of a Shifted Circle or Cardioid:
When the radius is defined by a sine or cosine function (optionally added to a constant), the graph results in a displaced circle or a cardioid:
Equation of a Rose Curve:
Using trigonometric functions with a multiple of the angle yields a flower-like pattern with multiple petals:
Equation of a Limacon:
Combining constants with sine or cosine results in a limaçon, which may have inner loops or dimples depending on parameters:
These polar forms offer a concise and visually intuitive method for graphing a wide variety of curves.
A grid with concentric circles and straight radial lines, all centered at a single point called the pole, defines a polar coordinate system.
Each point on this grid is identified by its distance from the pole and by the angle it makes with the polar axis.
A graph where every point stays the same distance from the origin forms a perfect circle. The radius remains constant throughout the rotation.
A straight line in polar coordinates is traced when the angle remains fixed and the radius increases.
When the radius is represented by a cosine function, the path traced by this radius forms a circle with the origin on its edge.
Changing the sign of the constant a flips the circle across the vertical axis.
The polar graphs help in tracing satellite orbits around Earth. Here, r represents the satellite's distance from the centre of the Earth, and θ is the angle measured from a reference line. This allows accurate tracking of the satellite, as its angle values change continuously, providing precise information about its position relative to Earth's surface.
From Chapter 8:
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8.2 : Graphs of Polar Equations
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