Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

Plot the points whose polar coordinates are \((3,2 \pi)\), \(\left(-2, \frac{1}{3} \pi\right),\left(-2,-\frac{1}{4} \pi\right),(-1,1),(1,-4 \pi),\left(\sqrt{3},-\frac{7}{6} \pi\right),\left(-2, \frac{1}{4} \pi\right)\), and \(\left(-1,-\frac{1}{2} \pi\right)\).

Step-by-Step Solution

Answer

Translate polar coordinates with negative radii or angles to standard form and plot on a polar graph.

Step 1: Understand Polar Coordinates

Polar coordinates consist of a radius, \( r \), and an angle, \( \theta \), which specifies a point in the plane using distance from the origin and an angle from the positive x-axis.

Step 2: Convert Negative Radius to Positive

For any points with negative \( r \), convert them to positive by adding \( \pi \) to the angle (\( \theta \)) and taking the absolute value of \( r \). For example, with \((-2, \frac{1}{3} \pi)\), convert to \((2, \frac{1}{3} \pi + \pi) = (2, \frac{4}{3} \pi)\).

Step 3: Adjust Angles As Needed

Ensure all angles are within \([0, 2\pi)\) by adding or subtracting \(2\pi\). For instance, \(-\frac{7}{6} \pi\) becomes \(2\pi - \frac{7}{6} \pi = \frac{5\pi}{6}\).

Step 4: Plot Points on Polar Plane

For each point in polar coordinates, position the point using the radius to determine the distance from the pole (origin) and the angle to determine the rotational position from the positive x-axis.

Step 5: Annotate the Plot

Label each plotted point on the polar graph with its polar coordinates for clarity.

Key Concepts

Radius and Angle

In polar coordinates, every point on a plane is described using two elements: the radius, denoted as \( r \), and the angle, denoted as \( \theta \). Think of the radius as the distance from the origin (also known as the pole) to the point. The angle indicates the rotation from the positive x-axis towards the point.

• The radius tells us how far the point is from the central origin. If \( r \) is positive, it means the point is directly in the direction of the angle \( \theta \).
• If \( r \) is negative, the point is in the opposite direction of the angle \( \theta \), similar to reflecting it across the origin.
• The angle \( \theta \) is measured in radians, providing a precise way to determine the point's rotation around the origin.

Understanding how these elements define a point in polar coordinates is crucial for effectively plotting and manipulating these points.

Coordinate Conversion

Converting polar coordinates that have a negative radius to a more standard form involves a simple adjustment. If you encounter a negative radius, like in the point \((-2, \frac{1}{3} \pi)\), you transform the coordinates using the following steps:

• Take the absolute value of \( r \). This changes \(-2\) to \(2\).
• Add or subtract \( \pi \) (which is \(180^\circ\)) to the angle \( \theta \) to account for the directional change.

For this example, after conversion, you will get \((2, \frac{4}{3} \pi)\). This step ensures that all negatives are turned into positives, allowing for a more intuitive understanding of the point's location.

Angle Adjustment

Keeping angles within a specific range is vital when working with polar coordinates. Typically, angles are adjusted to fall within the interval \([0, 2\pi)\). This ensures consistency, especially when visualizing coordinates or performing calculations.
If you have an angle like \(-\frac{7}{6}\pi\), you can adjust it by adding \(2\pi\) to make it positive, as follows:

• Calculate \(2\pi - \frac{7}{6}\pi\) to adjust the angle.
• The result, \( \frac{5\pi}{6} \), is the positive equivalent within the standard angle range.

By standardizing angles this way, you preserve uniformity across your coordinates, making them simpler to work with.

Plotting Points

Once you've prepared the polar coordinates, plotting them is straightforward. The process involves using the radius and angle to locate points on a graph. Here is a simple outline for plotting:

• Start from the origin or pole and position your point at a distance equal to the radius \( r \).
• From the positive x-axis, rotate by the angle \( \theta \).
• Draw a mark on a polar plot where these two instructions intersect.

Make sure to clearly label your plotted points with their polar coordinates for easy identification. This practice not only enhances visual clarity but also reinforces understanding of the conversion from polar coordinates to a graphical layout.