Can someone explain a polar coordinate conversion?

In summary, the process of changing (2x - x2)1/2 to 2 cos θ involves manipulating the entire equation and not just the expression itself. This is done by squaring both sides, substituting in polar coordinates, and dividing by r.
  • #1
sc5678
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I am having trouble understanding how (2x - x2)1/2 becomes 2 cos θ.

Thanks
 
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  • #2
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]".
 

FAQ: Can someone explain a polar coordinate conversion?

What are polar coordinates?

Polar coordinates are a system of representing points in a plane using a distance from a fixed point (known as the origin) and an angle from a fixed direction (known as the polar axis).

How do you convert Cartesian coordinates to polar coordinates?

To convert Cartesian coordinates (x,y) to polar coordinates (r,θ), use the following formulas: r = √(x² + y²) and θ = arctan(y/x). Keep in mind that θ is measured in radians, not degrees.

What is the purpose of using polar coordinates?

Polar coordinates are useful for representing points in a circular or curved shape, as they are based on distance and angle. They are commonly used in mathematics, physics, and engineering for applications such as graphing polar equations and analyzing circular motion.

Can polar coordinates be negative?

Yes, polar coordinates can be negative. The distance (r) can be negative if the point lies in the opposite direction of the polar axis, and the angle (θ) can be negative if the point is below the polar axis.

How do you convert polar coordinates to Cartesian coordinates?

To convert polar coordinates (r,θ) to Cartesian coordinates (x,y), use the following formulas: x = r cos(θ) and y = r sin(θ). Keep in mind that θ is measured in radians, not degrees.

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