Can someone explain a polar coordinate conversion?

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SUMMARY

The discussion clarifies the process of converting Cartesian coordinates to polar coordinates, specifically addressing the equation y = (2x - x^2)^{1/2}. The transformation requires manipulating the entire equation rather than altering individual expressions. The correct conversion leads to the equation r^2 = 2r cos(θ), which simplifies to r = 2 cos(θ) after dividing by r. This highlights the importance of handling the complete equation for accurate polar coordinate conversion.

PREREQUISITES
  • Understanding of Cartesian coordinates and polar coordinates
  • Familiarity with algebraic manipulation of equations
  • Knowledge of trigonometric functions, particularly cosine
  • Basic concepts of coordinate transformations in mathematics
NEXT STEPS
  • Study the process of converting equations from Cartesian to polar coordinates
  • Learn about the geometric interpretation of polar coordinates
  • Explore the use of trigonometric identities in coordinate transformations
  • Practice solving various equations using polar coordinate conversions
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Students studying mathematics, particularly those focusing on calculus and coordinate geometry, as well as educators teaching these concepts.

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I am having trouble understanding how (2x - x2)1/2 becomes 2 cos θ.

Thanks
 
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It doesn't. You can't just change a single expression into a particular coordinate system. In order to change to polar coordinates, you have to have an equation or function. Is this [itex]y= (2x- x^2)^{1/2}[/itex]? If so, then you can start by squaring both sides: [itex]y^2= 2x- x^2[/itex] so that [itex]x^2+ y^2= 2x[/itex].

Now, [itex]x^2+ y^2= r^2[/itex] and [itex]2x= 2r cos(\theta)[/itex]. The entire equation is now [itex]r^2= 2r cos(\theta)[/itex] and, dividing both sides by r, [itex]r= 2 cos(\theta)[/itex]. But notice that this was reached by manipulating the whole equation- it was not just "[itex](2x- x^2)^{1/2}[/itex]" that became "[itex]2 cos(\theta)[/itex]".
 

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