In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. _ (2,−2√3)

Verified step by step guidance

1. Recall that to convert rectangular coordinates \((x, y)\) to polar coordinates \((r, \theta)\), we use the formulas: \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\).
2. Calculate the radius \(r\) by substituting \(x = 2\) and \(y = -2\sqrt{3}\) into the formula: \(r = \sqrt{(2)^2 + (-2\sqrt{3})^2}\).
3. Simplify the expression under the square root to find the exact value of \(r\).
4. Find the angle \(\theta\) by calculating \(\theta = \tan^{-1}\left(\frac{-2\sqrt{3}}{2}\right)\), which simplifies to \(\tan^{-1}(-\sqrt{3})\).
5. Determine the correct quadrant for \(\theta\) based on the signs of \(x\) and \(y\) (here, \(x > 0\) and \(y < 0\)), and adjust \(\theta\) accordingly to express it in radians.

Key Concepts

Rectangular and Polar Coordinate Systems

Rectangular coordinates (x, y) represent a point's position using horizontal and vertical distances from the origin. Polar coordinates (r, θ) describe the same point by its distance r from the origin and the angle θ formed with the positive x-axis, measured in radians.

Conversion Formulas Between Coordinates

To convert from rectangular to polar coordinates, use r = √(x² + y²) to find the radius and θ = arctan(y/x) to find the angle. Adjust θ based on the quadrant of the point to ensure the angle correctly represents the point's direction.

Angle Measurement in Radians and Quadrant Considerations

Angles in polar coordinates are measured in radians, where 2π radians equal 360°. Since arctan(y/x) returns values between -π/2 and π/2, you must consider the signs of x and y to determine the correct quadrant and adjust θ accordingly for an accurate angle.