Problem 25 - Precalculus with Calculus Previews

Question

Find polar coordinates that satisfy (a) \(r>0,-\pi<\theta \leq \pi\) (b) \(r<0,-\pi<\theta \leq \pi\) for each point with the given rectangular coordinates. $$ (-2,-2) $$

Step-by-Step Solution

Answer

(a) \((2\sqrt{2}, -\frac{3\pi}{4})\); (b) \((-2\sqrt{2}, \frac{\pi}{4})\)

Step 1: Convert Rectangular to Polar Coordinates

Given rectangular coordinates (-2, -2), we first calculate the value of \( r \) using the formula for polar coordinates: \( r = \sqrt{x^2 + y^2} \). This becomes \( r = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \).

Step 2: Find Primary Angle Θ

Using the formula \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \), we calculate \( \theta = \tan^{-1}\left(\frac{-2}{-2}\right) = \tan^{-1}(1) \). Since both x and y are negative, the point is in quadrant III, yielding \( \theta = -\frac{3\pi}{4} \).

Step 3: Adjust θ for Positive r

Since \( r > 0 \) already, we can keep \( r = 2\sqrt{2} \). Our angle \( \theta = -\frac{3\pi}{4} \) is within the given range \( -\pi < \theta \leq \pi \), so no adjustment is needed. The polar coordinates for condition (a) are \( (r, \theta) = (2\sqrt{2}, -\frac{3\pi}{4}) \).

Step 4: Adjust θ for Negative r

For \( r < 0 \), we set \( r = -2\sqrt{2} \). To modify \( \theta \) such that \( r < 0 \), we must add \(\pi\) to angle to have it represent the direction opposite to its current. Recalculate \( \theta = -\frac{3\pi}{4} + \pi = \frac{\pi}{4} \), which is in quadrant I. So the polar coordinates for condition (b) are \( (r, \theta) = (-2\sqrt{2}, \frac{\pi}{4}) \).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, often referred to as Cartesian coordinates, are a way to specify the location of a point in a plane using two numbers. These numbers correspond to distances along two perpendicular axes, typically labeled as the x-axis and y-axis. Each point in the plane can be uniquely identified by an ordered pair of numbers, written as \(x, y\). In the context of the given problem, the point \((-2, -2)\)\ indicates that from the origin (0,0), the point is 2 units in the negative x-direction and 2 units in the negative y-direction.

Understanding the conversion from rectangular to polar coordinates, however, provides a different, often more insightful picture, especially when dealing with circular or rotational motion.

Quadrant

A quadrant is one of the four sections of the Cartesian plane. The quadrants are numbered counter-clockwise starting from the positive x and y-axis section. They are:

• Quadrant I: Both x and y are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive, y is negative.

Understanding the quadrant in which a point lies is crucial when converting to polar coordinates, particularly in determining the angle. In the given problem, the coordinates (-2, -2) locate the point in Quadrant III. This is determined by the fact that both the x and y values are negative. The angle calculation must take this quadrant into account, affecting how the angle is computed and interpreted.

Angle Conversion

Angle conversion is an essential step in transitioning from rectangular to polar coordinates. Here, we express the position of a point not just by distance from the origin, but also by an angle from a reference direction. The most common angle measurement used in mathematics is in radians. To find the angle \(\theta\), use the formula \(\theta = \tan^{-1}(\frac{y}{x})\).

In our problem, with coordinates (-2, -2), this results in an angle of \(\tan^{-1}(1) \), or \(-\frac{3\pi}{4}\). This angle reflects the point's position relative to the negative x-axis in Quadrant III.

Finally, when shifting angles to accommodate a change in the sign of the radius, such as converting to a negative radius for polar coordinates, adding \(\pi\) to the angle effectively moves the direction to its opposite. This ensures the polar representation remains correct and consistent with the properties of the coordinates.