Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.  


a. The point with Cartesian coordinates (−2, 2) has polar coordinates (2√2, 3π/4), (2√2, 11π/4), (2√2, −5π/4), and (−2√2,−π/4).  

Verified step by step guidance

1. Recall the relationship between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\): \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\), where \(r = \sqrt{x^2 + y^2}\) and \(\theta = \arctan\left(\frac{y}{x}\right)\) (adjusted for the correct quadrant).
2. Calculate the radius \(r\) for the point \((-2, 2)\): \(r = \sqrt{(-2)^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}\).
3. Determine the angle \(\theta\): Since \(x = -2\) and \(y = 2\), the point lies in the second quadrant. The reference angle is \(\arctan\left(\frac{2}{-2}\right) = \arctan(-1)\), but we must add \(\pi\) to get the correct angle in the second quadrant, so \(\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\).
4. Check each given polar coordinate to see if it corresponds to the Cartesian point \((-2, 2)\) by converting back to Cartesian using \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\):
5. - For \((2\sqrt{2}, \frac{3\pi}{4})\), verify if \(x = 2\sqrt{2} \cos(\frac{3\pi}{4})\) and \(y = 2\sqrt{2} \sin(\frac{3\pi}{4})\) equal \((-2, 2)\).
6. - Repeat this for \((2\sqrt{2}, \frac{11\pi}{4})\), \((2\sqrt{2}, -\frac{5\pi}{4})\), and \((-2\sqrt{2}, -\frac{\pi}{4})\).
7. Remember that polar coordinates are not unique: adding or subtracting multiples of \(2\pi\) to \(\theta\) gives the same point, and using a negative \(r\) with \(\theta\) shifted by \(\pi\) also represents the same point. Use these facts to explain which of the given coordinates are valid representations of \((-2, 2)\).

Key Concepts

Conversion between Cartesian and Polar Coordinates

This concept involves translating a point from Cartesian coordinates (x, y) to polar coordinates (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. The formulas are r = √(x² + y²) and θ = arctan(y/x), adjusted for the correct quadrant.

Multiple Representations of Polar Coordinates

A single point can have infinitely many polar coordinates because adding or subtracting multiples of 2π to the angle θ results in the same direction. Also, negative radius values with adjusted angles can represent the same point, reflecting the non-uniqueness of polar coordinates.

Quadrant Considerations in Angle Determination

When converting to polar form, the angle θ must be placed in the correct quadrant based on the signs of x and y. This ensures the angle accurately represents the point's direction, as arctan alone only gives values between -π/2 and π/2.