Convert the Cartesian equation to the polar equation

In summary, converting a Cartesian equation to a polar equation allows us to represent a mathematical relationship between variables in a different coordinate system, which can be useful for dealing with circular or symmetrical shapes. To convert a Cartesian equation to a polar equation, we use specific formulas based on the distance from the origin and the angle from the positive x-axis. Some common Cartesian equations and their corresponding polar equations include linear equations, circles, and ellipses. The benefits of using a polar equation over a Cartesian equation include simplifying the representation of shapes and aiding in solving problems involving symmetry. However, limitations of using polar equations include only being applicable in two-dimensional space and potential difficulty in visualization and conversion.
  • #1
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Homework Statement


Replace the Cartesian equation with an equivalent polar equation.
##x^2 + (y - 18)^2 = 324##
a)##r = 36 sin θ##
b)##r^2 = 36 cos θ##
c)##r = 18 sin θ##
d)##r = 36 cos θ##

Homework Equations


##x= r cos \theta ##
##y= r sin \theta ##
##x^2 + y^2 = r^2 ##

The Attempt at a Solution


##x^2 + (y - 18)^2 = 324##
##x^2 + y^2 - 2*18*y + 18^2 = 324 ##
##x^2 + y^2 - 36 y =0##
##r^2 = 36 y##
Divide by r :
##r = 36 r sin \theta##
So , the answer is (a) .Right ?
 
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  • #2
Your derivation looks right to me.
 
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Likes Fatima Hasan

1. What is the purpose of converting a Cartesian equation to a polar equation?

Converting a Cartesian equation to a polar equation allows us to represent a mathematical relationship between variables in a different coordinate system. This can be useful in certain situations, such as when dealing with circular or symmetrical shapes.

2. How do you convert a Cartesian equation to a polar equation?

To convert a Cartesian equation to a polar equation, we use the following formulas:
x = r * cos(theta)
y = r * sin(theta)
r = sqrt(x^2 + y^2)
theta = arctan(y/x)
where r represents the distance from the origin and theta represents the angle from the positive x-axis.

3. What are some common Cartesian equations and their corresponding polar equations?

Some common Cartesian equations and their corresponding polar equations are:
- y = mx + b (linear equation)
polar equation: r = -b/sin(theta - m)
- x^2 + y^2 = r^2 (circle)
polar equation: r = a
- x^2/a^2 + y^2/b^2 = 1 (ellipse)
polar equation: r = (a*b)/sqrt(b^2*cos^2(theta) + a^2*sin^2(theta))

4. What are the benefits of using a polar equation over a Cartesian equation?

One benefit of using a polar equation is that it can simplify the representation of certain shapes, such as circles and ellipses. Additionally, polar equations can be useful in solving problems involving symmetry, as the polar coordinate system is based on symmetry around a central point. They can also be used to describe motion in a circular or spiral pattern.

5. Are there any limitations to using polar equations?

One limitation of using polar equations is that they can only be used to describe relationships between variables in two-dimensional space. They also may not be as intuitive or easy to visualize as Cartesian equations for some people. Additionally, converting between polar and Cartesian coordinates can sometimes be complicated and time-consuming.

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