Problem 24 - Algebra and Trigonometry

Question

Plot each point given in polar coordinates. $$ (-3, \pi) $$

Step-by-Step Solution

Answer

Plot the point at (3, 0).

Step 1: Understand Polar Coordinates

Polar coordinates \(r, \theta\) are described by a radius \(r\) and an angle \(\theta\). The radius \(r\) represents the distance from the origin, and \(\theta\) is the counterclockwise angle from the positive x-axis.

Step 2: Identify the Given Coordinates

The given coordinates are \(-3, \pi\). Here, \r = -3\ and \theta = \pi\ (or 180°).

Step 3: Convert the Negative Radius

A negative radius indicates that the point is in the direction opposite to the angle \(\theta\). So, convert the coordinates \(-3, \pi\) to \left(3, \pi + \pi\right) = (3, 2\pi)\. Since \2\pi\ is equivalent to 0 radians, we can simplify this to \(3, 0\).

Step 4: Plot the Point

Plot the point \(3, 0\) on the polar coordinate system. This means moving 3 units from the origin at an angle of 0 radians (along the positive x-axis).

Key Concepts

Radius

In polar coordinates, the **radius** is denoted as \(r\). This value represents the distance from the origin (the center of the coordinate system) to the point in question. A positive radius means the point is in the standard direction from the origin at an angle \(\theta\). However, if the radius is **negative**, it indicates that the point is in the opposite direction to the given angle. For instance, if you have coordinates \((-3, \pi)\), the radius is -3, meaning you should move 3 units opposite to the angle of \(\pi\).

Here are key points to remember:

• Radius is the distance from the origin to the point.
• Positive radius means normal direction.
• Negative radius means opposite direction.
• Always interpret the sign of radial distance carefully.

Understanding radius correctly helps in accurately rendering polar coordinates in the plane.

Angle in Radians

The **angle in radians**, denoted as \(\theta\) in polar coordinates, is the measure of the counter-clockwise angle from the positive x-axis to the point. Unlike degrees, radians provide a mathematically convenient measure for many calculations, particularly in trigonometry and calculus areas. One full circle equals \(2\pi\) radians.

Here's a quick guide:

• 0 radians is the positive x-axis.
• \(\pi/2\) radians is the positive y-axis.
• \(\pi\) radians is the negative x-axis.
• \(3\pi/2\) radians is the negative y-axis.
• \(2\pi\) radians completes a full circle back to the positive x-axis.

For example, in the given coordinates \((-3, \pi)\), \(\pi\) radians corresponds to the angle of 180 degrees, which places the point on the negative x-axis. By understanding angles in radians, you can accurately locate the direction in which to plot the radius.

Plotting Points

To **plot points** in a polar coordinate system, you combine the concepts of radius and angle in radians. Start at the origin, move outward by the radius distance, and turn according to the angle's measure in radians. If the radius is negative, move in the direction opposite to that of the angle.

Following steps for \((-3, \pi)\):

• Identify the radius (-3) and the angle (\(\pi\)).
• Convert the negative radius: \((-3, \pi)\) becomes \((3, \pi + \pi) = (3, 2\pi)\).
• Simplify to an equivalent angle: \((3, 2\pi)\) simplifies to \((3, 0)\).
• Plot the point by moving 3 units from the origin at 0 radians, along the positive x-axis.

These steps ensure accurate plotting of the point according to the given polar coordinates.