Finding a function in x,y from function in polar coordinates

In summary, v is in polar coordinates and i want to fin u(x,y) knowing that v(r,theta)=u(rcos(theta),rsin(theta)). By applying double-angle formulas and substituting \theta=\tan^{-1}(y/x), we can simplify u(x,y)=9+18cos(2arctan(y/x))-9sin(4arctan(y/x)) into algebraic expressions.
  • #1
sara_87
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Homework Statement


v is in polar coordinates and i want to fin u(x,y) knowing that v(r,theta)=u(rcos(theta),rsin(theta))
therefore, u(x,y)=v(sqrt(x^2+y^2), arctan(y/x))
v(r,theta) = 9+18cos(2(theta))-9sin(4(theta))
question: what is u(x,y)?

Homework Equations





The Attempt at a Solution



u(x,y)=9+18cos(2arctan(y/x))-9sin(4arctan(y/x))

Is this correct and can i simplify this more?
Thank you.
 
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  • #2


It's correct, but you can simplify a lot more. I would use double-angle formulas to express [itex]\sin(4\theta)[/itex] and [itex]\cos(2\theta)[/itex] in terms of [itex]\sin(\theta)[/itex] and [itex]\cos(\theta)[/itex]. Then I could substitute [itex]\theta=\tan^{-1}(y/x)[/itex] and work the resulting expressions into algebraic expressions.
 

1. How do you convert a function from polar coordinates to Cartesian coordinates?

To convert a function from polar coordinates to Cartesian coordinates, you can use the following formulas:

x = r * cos(theta) and y = r * sin(theta)

where r is the distance from the origin and theta is the angle measured counterclockwise from the positive x-axis.

2. Can any function in polar coordinates be expressed as a function in Cartesian coordinates?

Yes, any function in polar coordinates can be expressed as a function in Cartesian coordinates. However, the reverse is not always true as some functions in Cartesian coordinates may not have an equivalent representation in polar coordinates.

3. How do you graph a function in polar coordinates?

To graph a function in polar coordinates, you can plot points on a polar grid using the values of r and theta. You can also use a graphing calculator or software to visualize the graph of the function.

4. What is the relationship between the graphs of a function in polar and Cartesian coordinates?

The graph of a function in polar coordinates is a spiral-like curve, while the graph of the same function in Cartesian coordinates is a line or a curve. The two graphs are related by the conversion formulas mentioned in the first question.

5. How do you find the domain and range of a function in polar coordinates?

The domain of a function in polar coordinates is the set of all possible values of theta, while the range is the set of all possible values of r. The domain and range can be determined by analyzing the behavior of the function as theta approaches certain values, such as 0, pi/2, pi, etc.

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