Problem 32 - 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. $$ (1,-1) $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Relationship between Rectangular and Polar Coordinates

Rectangular coordinates \((x, y)\) can be converted to polar coordinates \((r, \theta)\) using the formulas:\[ r = \sqrt{x^2 + y^2} \]\[ \theta = \text{atan2}(y, x) \]This step helps to convert the given rectangular coordinates into polar form.

Step 2: Calculate the Radius \(r\)

For the point \((1, -1)\), calculate the radius \(r\):\[ r = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

Step 3: Calculate the Angle \(\theta\)

Use the \(\text{atan2}\) function to find \(\theta\) for the point \((1, -1)\):\[ \theta = \text{atan2}(-1, 1) = -\frac{\pi}{4} \]This angle falls within the range \(-\pi < \theta \leq \pi\).

Step 4: Find Polar Coordinates with \(r > 0\)

For the condition \(r > 0\):The polar coordinates are \( (\sqrt{2}, -\frac{\pi}{4}) \) because \(r = \sqrt{2} > 0\) and \(\theta = -\frac{\pi}{4}\) satisfies \(-\pi < \theta \leq \pi\).

Step 5: Find Polar Coordinates with \(r < 0\) and Adjust \(\theta\)

For the condition \(r < 0\), you need to adjust \(\theta\) because \(r\) must be negative. Adjust \(\theta\) as follows:\[ r = -\sqrt{2} \]To maintain the direction, add \(\pi\) to the original \(\theta\):\[ \theta = \theta + \pi = -\frac{\pi}{4} + \pi = \frac{3\pi}{4} \]Thus, the polar coordinates are \((-\sqrt{2}, \frac{3\pi}{4})\).

Key Concepts

Rectangular Coordinates

When we talk about rectangular coordinates, we are referring to the common Cartesian coordinate system. This system uses two values, usually denoted as \((x, y)\), to determine the position of a point on a plane.

• The "x" coordinate is the horizontal distance from the origin to the point.
• The "y" coordinate is the vertical distance from the origin to the point.

For example, the point \((1, -1)\), means the point is positioned one unit to the right of the origin and one unit down. Rectangular coordinates are useful for plotting graphs and performing many types of calculations, easily conveying the relative position in a straightforward manner. Understanding how to interpret these coordinates is essential before proceeding to convert them into other forms, such as polar coordinates.

Coordinate Conversion

Coordinate conversion refers to the process of transforming one set of coordinates into another. In our case, converting rectangular coordinates \((x, y)\) into polar coordinates \((r, \theta)\) involves both mathematical intuition and understanding formulas.The conversion steps are:

• Calculate the radius \(r\) using the formula: \[ r = \sqrt{x^2 + y^2} \] This gives you the distance of the point from the origin.
• Determine the angle \(\theta\) using the trigonometric function \(\text{atan2}(y, x)\): \[ \theta = \text{atan2}(y, x) \] This finds the angle the line to the point makes with the positive x-axis.

For our specific example of the point \((1, -1)\), the conversion results in:

• Radius: \(r = \sqrt{2}\)
• Angle: \(\theta = -\frac{\pi}{4}\)

Different scenarios may require adjustments to the angle to meet specific criteria, such as ensuring the radius is negative or positive as needed.

Polar Coordinate System

The polar coordinate system provides a unique way to represent points on a plane, distinct from rectangular coordinates. In this system, a point is described by:

• \(r\), the distance from the point to the origin (akin to the radius of a circle), and
• \(\theta\), the angle formed by the point, the origin, and the positive x-axis.

In polar coordinates, angles are usually measured in radians. A key aspect of this system is that the same point can have multiple representations. For instance:

• With \(r > 0\), the angle \(\theta\) can be as straightforward as the one calculated using \(\text{atan2}\), such as \((\sqrt{2}, -\frac{\pi}{4})\).
• For \(r < 0\), the angle must be adjusted by adding \(\pi\) to ensure correct direction, resulting in \((-\sqrt{2}, \frac{3\pi}{4})\).

Understanding how to work within this coordinate system is valuable, as it provides perspective that is often more useful in contexts involving circular motion or where angles and distances from a central point are more relevant than linear distances.