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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblema 12.2.77a
Capitolo 12, Problema 12.2.77a

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).  

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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).
Calculate the radius \(r\) for the point \((-2, 2)\): \(r = \sqrt{(-2)^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}\).
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 \(\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\).
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 \(y = r \sin(\theta)\):
- 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 \((-2, 2)\).
- Repeat this for \((2\sqrt{2}, \frac{11\pi}{4})\), \((2\sqrt{2}, -\frac{5\pi}{4})\), and \((-2\sqrt{2}, -\frac{\pi}{4})\).
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 \((-2, 2)\).

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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.
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Intro to Polar Coordinates

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.
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Intro to 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.
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Trig Values in Quadrants II, III, & IV
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