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 \(\left(-1,-\frac{1}{2} \pi\right) .\)

Short answer

Plot each point as directed by adjusting for negative radial distances and angles.

Step-by-step solution

Understanding Polar Coordinates

Polar coordinates are represented as \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle measured from the positive x-axis in a counter-clockwise direction. If \(r\) is negative, the direction is opposite to what \(\theta\) indicates.

Plot Point \((3, 2\pi)\)

This point corresponds to \(r = 3\), \(\theta = 2\pi\). Since \(2\pi\) means a full rotation, this point lies on the positive x-axis, 3 units away from the origin.

Plot Point \((-2, \frac{1}{3}\pi)\)

For \(r = -2\), \(\theta = \frac{1}{3}\pi\), calculate the position: \(\theta = \frac{1}{3}\pi\) points 60° from the positive x-axis. Because \(r\) is negative, move 2 units in the opposite direction, so plot in the direction 60° + 180°.

Plot Point \((-2, -\frac{1}{4}\pi)\)

Here, \(r = -2\), \(\theta = -\frac{1}{4}\pi\) (clockwise). Convert to positive, which results in \(\theta = -\frac{1}{4}\pi + \pi = \frac{3}{4}\pi\), then shift 2 units in the opposite direction.

Plot Point \((-1, 1)\)

First, find the equivalent angle by converting \(1\) radian to degrees or noting it is a little less than \(\frac{1}{3}\pi\). Then plot 1 unit in the opposite direction due to negative \(r\).

Plot Point \((1, -4\pi)\)

Multiply \( -4\pi \) by 180/\(\pi\) to convert, which is -720°. Subtract 360° twice to get 0°, then plot 1 unit on the x-axis as \(r\) is positive.

Plot Point \((\sqrt{3}, -\frac{7}{6}\pi)\)

\( -\frac{7}{6}\pi \) converts to negative angles, reduce by adding \(2\pi\) to find equivalent angle, position is just past 210°, and move \(\sqrt{3}\) units.

Plot Point \((-2, \frac{1}{4}\pi)\)

Convert equivalent to 45°, adjust \(r\) to move opposite direction due to negative \(r\).

Plot Point \((-1, -\frac{1}{2}\pi)\)

Convert by adjusting the angle to a positive equivalent, 270°. Plot one unit in the opposite for negative \(r\).

Key concepts

Radial Distance

In polar coordinates, radial distance is denoted by the symbol \( r \). This represents the distance from the origin, which is the center of the coordinate system. Unlike the Cartesian system, which uses x and y coordinates to plot positions, the radial distance in polar coordinates extends outwards from this central point.

• For a positive \( r \), the point is located in the direction of the angle \( \theta \).
• If \( r \) is negative, the point is placed in the opposite direction of \( \theta \).

Understanding radial distance is crucial as it dictates how far from the origin your point will be. It's helpful to visualize this as a line segment between the origin and the plotted point.

Angle Measurement

Angle measurement in polar coordinates, represented as \( \theta \), is crucial for determining the direction of the point from the origin. Typically, angles are measured in radians from the positive x-axis.

• Counter-clockwise movements correspond to positive angles.
• Conversely, clockwise movements correspond to negative angles.

This circular measurement allows for a complete rotation, either positive or negative, around the origin. In some problems, you'll find angles greater than \( 2\pi \) or less than \(-2\pi\). It's normal to convert such angles to their equivalent within the standard circle by adding or subtracting \( 2\pi \).
Remember, understanding angle measurement helps you properly locate the direction in which you measure the radial distance.

Conversion of Angles

Sometimes you need to convert between degrees and radians, especially when plotting points or interpreting angles. For conversion:

• 1 radian = 180/\( \pi \) degrees
• 1 degree = \( \pi \)/180 radians

Besides changing units, converting angles is necessary when simplifying expressions. When you have a large positive or negative angle, reducing it by adding or subtracting full rotations (i.e., multiples of \( 2\pi \)) can make graphing and understanding easier.
This adjustment helps align the angle within the typical scope of \( 0 \) to \( 2\pi \), which simplifies location and visualization on a polar graph.

Graphing Technique

Plotting points in polar coordinates can seem tricky at first. However, it becomes easier with understanding and practice. Here’s how you can graph these points:

• First, determine if the radial distance \( r \) is positive or negative to decide the direction of your plotting from the origin.
• Measure the angle \( \theta \) from the positive x-axis, accounting for whether it’s positive or negative, by rotating counter-clockwise or clockwise respectively.
• If you encounter a complex angle, convert it to an equivalent location by rotations of \( 2\pi \).

The graphing technique ensures that each plotted point accurately reflects the coordinate pair given in polar terms. Over time and with practice, this approach aids in visualizing and understanding geometric relationships within the circle of the polar coordinate system.