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

(1, √3)

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Recall the formulas to convert Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\): \(r = \sqrt{x^2 + y^2}\) \(\theta = \arctan\left(\frac{y}{x}\right)\)

Calculate the radius \(r\) by substituting \(x = 1\) and \(y = \sqrt{3}\) into the formula: \(r = \sqrt{1^2 + (\sqrt{3})^2}\)

Calculate the angle \(\theta\) using the arctangent formula: \(\theta = \arctan\left(\frac{\sqrt{3}}{1}\right)\)

Express the polar coordinates as \((r, \theta)\) using the values found in steps 2 and 3.

For the second way, express the angle \(\theta\) in degrees instead of radians or use a known angle value (e.g., \(\theta = \frac{\pi}{3}\) radians or \(60^\circ\)) to write the polar coordinates 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 their distance from the origin (r) and the 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. Adjust θ based on the quadrant to get the correct direction.

Multiple Representations of Polar Coordinates

Polar coordinates are not unique; the same point can be represented by adding multiples of 2π to θ or by using negative r values with adjusted angles. Recognizing these alternatives allows expressing coordinates in at least two different valid ways.

31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.(1, √3)