Rewrite the polar equation
as a Cartesian equation.
The goal is to eliminate
and
and introduce
and
We clear the fraction, and then use substitution. In order to replace
with
and
we must use the expression
The Cartesian equation is
However, to graph it, especially using a graphing calculator or computer program, we want to isolate
When our entire equation has been changed from
and
to
and
we can stop, unless asked to solve for
or simplify. See
[link] .
The “hour-glass” shape of the graph is called a
hyperbola . Hyperbolas have many interesting geometric features and applications, which we will investigate further in
Analytic Geometry .
The polar grid is represented as a series of concentric circles radiating out from the pole, or origin.
To plot a point in the form
move in a counterclockwise direction from the polar axis by an angle of
and then extend a directed line segment from the pole the length of
in the direction of
If
is negative, move in a clockwise direction, and extend a directed line segment the length of
in the direction of
See
[link] .
If
is negative, extend the directed line segment in the opposite direction of
See
[link] .
To convert from polar coordinates to rectangular coordinates, use the formulas
and
See
[link] and
[link] .
To convert from rectangular coordinates to polar coordinates, use one or more of the formulas:
and
See
[link] .
Transforming equations between polar and rectangular forms means making the appropriate substitutions based on the available formulas, together with algebraic manipulations. See
[link] ,
[link] , and
[link] .
Using the appropriate substitutions makes it possible to rewrite a polar equation as a rectangular equation, and then graph it in the rectangular plane. See
[link] ,
[link] , and
[link] .
Section exercises
Verbal
How are polar coordinates different from rectangular coordinates?
For polar coordinates, the point in the plane depends on the angle from the positive
x- axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations.
the transfer of energy by a force that causes an object to be displaced; the product of the component of the force in the direction of the displacement and the magnitude of the displacement
A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.
A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike
Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.
the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon