31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.

(-4, 4√3)

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Recall the formulas to convert Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\): \(r = \sqrt{x^2 + y^2}\) and \(\theta = \arctan\left(\frac{y}{x}\right)\), but be careful with the quadrant of the point when determining \(\theta\).

Calculate the radius \(r\) using the given coordinates \(x = -4\) and \(y = 4\sqrt{3}\): \(r = \sqrt{(-4)^2 + (4\sqrt{3})^2}\).

Find the angle \(\theta\) by computing \(\arctan\left(\frac{y}{x}\right) = \arctan\left(\frac{4\sqrt{3}}{-4}\right)\), then adjust \(\theta\) to the correct quadrant since \(x\) is negative and \(y\) is positive (which places the point in the second quadrant).

Express the polar coordinates as \((r, \theta)\) where \(r\) is the radius found in step 2 and \(\theta\) is the angle found in step 3, typically in radians.

For the second method, use the fact that \(\theta\) can also be expressed as \(\pi - |\arctan(\frac{y}{|x|})|\) to directly find the angle in the second quadrant, then write the polar coordinates again as \((r, \theta)\).

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Cartesian and Polar Coordinate Systems

Cartesian coordinates represent points using (x, y) values on perpendicular axes, while polar coordinates describe points by a distance from the origin (r) and an angle (θ) from the positive x-axis. Understanding both systems is essential for converting between them.

Conversion Formulas Between Cartesian and Polar Coordinates

To convert from Cartesian (x, y) to polar (r, θ), use r = √(x² + y²) to find the radius and θ = arctan(y/x) to find the angle. Adjustments to θ may be needed depending on the quadrant where the point lies.

Multiple Representations of Polar Coordinates

Polar coordinates are not unique; the same point can be represented by different pairs (r, θ) by adding or subtracting full rotations (2π) to the angle or by using negative radius values with adjusted angles. Recognizing these variations helps express coordinates in multiple valid ways.

31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.(-4, 4√3)