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.
Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ...
Step 2: Find each score's deviation from the mean. ...
Step 3: Square each deviation from the mean. ...
Step 4: Find the sum of squares. ...
Step 5: Divide the sum of squares by n – 1 or N.
The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.
Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400
a. what is the probability of getting more than 12,000 hits?
b. what is the probability of getting fewer than 9,000 hits?
Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400.
a. What is the probability of getting more than 12,000 hits