Problem 33 - Algebra and Trigonometry

Question

Plot each point given in polar coordinates. $$ (-2,-\pi) $$

Step-by-Step Solution

Answer

Plot (2, \pi), 2 units left on the x-axis.

Step 1 – Understand Polar Coordinates

Polar coordinates are defined as \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle in radians measured from the positive x-axis.

Step 2 – Interpret Given Coordinates

The given coordinates are \((-2,-\pi)\). Here, \(-2\) is the radial distance and \(-\pi\) is the angle.

Step 3 – Convert to Equivalent Positive Radius

Since radius \(r\) is negative, we convert it to a positive value by adding \pi\ radians to the angle. Thus, \((-2, -\pi) = (2, \pi)\).

Step 4 – Plot the Point in Polar Coordinates

Plot the point \((2, \pi)\). This means the point is 2 units away from the origin in the direction of \(\pi\) radians, which is 180 degrees (or directly to the left on the x-axis).

Key Concepts

headline of the respective core concept

In polar coordinates, the radial distance is represented as \(r\).
It measures how far a point is from the origin (center) of the coordinate system.
The radial distance can be positive, which places the point in the direction of the angle,
or negative, which places it in the opposite direction.
For example, in the coordinates \((-2, -\pi)\), the radial distance is \(-2\),
indicating the point is 2 units away from the origin but in the opposite direction of the angle \(-\pi\).
Thus, understanding \(r\) helps in correctly positioning any point in the polar coordinate system.

headline of the respective core concept

An angle in radians is a way to measure angles where the distance around the circle's edge (arc length)
is proportionate to the radius.
\(\pi\) radians correspond to 180 degrees, and \(2\pi\) radians make a full circle of 360 degrees.
In polar coordinates, angles can also be negative,
meaning they go clockwise instead of the usual counterclockwise direction.
For the given point \((-2,-\pi)\), translating \(-\pi\) gives us an angle measured clockwise as 180 degrees
from the positive x-axis.
Converting to a more standard form, we switch the negativity by adjusting the direction of the radial distance.

headline of the respective core concept

Plotting points in polar coordinates requires both radial distance and angle measurements.
First, start from the origin (the center of the graph).
Determine the angle in radians and move in that direction.
This angle is typically measured counter-clockwise from the positive x-axis.
Then, from this direction, move outward by the radial distance defined in the coordinates.
For the exercise, after converting the point \((-2, -\pi) = (2,\pi)\),
start from the origin, align in the direction of \(\pi\) radians (or 180 degrees), and move 2 units to plot the point.
This visualizes the point's exact position on the polar graph, accurately reflecting its coordinates.