Converting Cartesian to Polar Coordinates: Explained with Example

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To convert Cartesian coordinates (-3, 4) to polar coordinates, the radius r is calculated as 5 using the formula r = sqrt(x² + y²). The angle θ is determined by θ = π - arctan(4/3) because the point lies in the second quadrant, where the cosine is negative and sine is positive. The range of arctan is from -π/2 to π/2, which necessitates adjusting the angle by subtracting from π to ensure it reflects the correct quadrant. The final polar coordinates are expressed as (5, π - arctan(4/3)). Understanding the quadrant placement is crucial for accurate conversion between coordinate systems.
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Homework Statement


Find polar coordinates.


Homework Equations


Cartesian: (-3,4)


The Attempt at a Solution


r = sqrt(9+16) = 5
sinθ = 4/5
cosθ = -3/5

θ = ∏ - arctan(4/3)

Answer: (5, ∏ - arctan(4/3))


I do not understand why we have subtracted the value arctan(4/3) from pi?
 
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What is the range of arctan? And where is the point really located?
 
Because the Cartesian coordinates (-3,4) are in 2th Quadrant
 

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