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What is the conversion formula from Cartesian to polar coordinates?

The conversion formula from Cartesian to polar coordinates is:

r = √(x² + y²)
θ = tan⁻¹(y/x)

To convert from Cartesian coordinates (x, y) to polar coordinates (r, θ), we use the following formulas:

r = √(x² + y²)
θ = tan⁻¹(y/x)

The first formula calculates the distance from the origin to the point (x, y) using the Pythagorean theorem. The second formula calculates the angle between the positive x-axis and the line connecting the origin to the point (x, y) using the inverse tangent function.

To see how these formulas work, consider the point (3, 4) in Cartesian coordinates. To convert to polar coordinates, we first calculate the distance from the origin:

r = √(3² + 4²) = √25 = 5

Next, we calculate the angle between the positive x-axis and the line connecting the origin to the point (3, 4):

θ = tan⁻¹(4/3) ≈ 0.93 radians ≈ 53.13 degrees

Therefore, the polar coordinates of the point (3, 4) are (5, 0.93).

It is important to note that the angle θ is measured in radians, not degrees. To convert from radians to degrees, we multiply by 180/π. To convert from degrees to radians, we multiply by π/180.

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