Problem 6 - Contemporary Precalculus

Question

Plot the point whose polar coordinates are given. $$(3, \pi / 6)$$

Step-by-Step Solution

Answer

Answer: To plot the point with polar coordinates (3, π/6), first draw the polar coordinate system. Then, locate the angle π/6 (30°) from the positive x-axis, and measure 3 units from the origin along the line that forms the angle π/6. The resulting point is the point with polar coordinates (3, π/6).

Step 1: Draw the polar coordinate system

To draw the polar coordinate system, draw a horizontal line (the x-axis) and a vertical line (the y-axis) intersecting at a point, which is the origin or the pole (denoted as O).

Step 2: Locate the angle \(\pi/6\)

The angle given is \(\pi/6\) radians, which is equivalent to \(30^\circ\) as 1 radian = \(180^\circ/\pi\). To find the angle, start at the positive x-axis and move counter-clockwise around the origin. Using a protractor, draw a line that forms an angle of \(30^\circ\) with the positive x-axis. Mark this angle as \(\pi/6\) or \(30^\circ\).

Step 3: Measure the distance from the origin

The distance given is 3 units. To plot the point, measure 3 units along the line that forms the angle \(\pi/6\) with the positive x-axis, starting from the origin O. Mark the point where the distance is 3 units, and label the point as P(3, \(\pi/6\)). This is the point with polar coordinates \((3, \pi/6)\).

Key Concepts

Polar Coordinate System

When we talk about the polar coordinate system, we refer to a two-dimensional coordinate system where each point on a plane is determined by a distance from a reference point and an angle from a reference direction. Unlike the Cartesian coordinate system, which uses a grid of horizontal and vertical lines, the polar coordinate system is based on radial distances and angles.
In the polar coordinate system, the reference point is called the pole, often represented as the origin in a Cartesian system. The reference direction is typically the positive x-axis, from which we measure angles. Angles can be measured in radians or degrees and start from the reference direction going counter-clockwise. Coordinates are expressed as ordered pairs \(r, \theta\), where \(r\) represents the radial distance from the pole and \(\theta\) represents the angular component measured from the reference direction. To plot the point \(3, \pi / 6\) as provided in the original exercise, you begin by drawing a polar grid, locate the angle of \(\pi / 6\) or 30 degrees from the positive x-axis, and then move radially outward from the pole to a distance of 3 units to mark the point.

Radians to Degrees Conversion

Understanding how to convert radians to degrees is crucial for working with angles in different contexts. Since radians and degrees both measure angles, being able to convert between the two allows for flexible use and interpretation of angular measurements.
The conversion is based on the relationship that \(180^\circ\) equals \(\pi\) radians. To convert radians to degrees, you multiply the radian measure by \(180^\circ/\pi\). Conversely, to convert degrees to radians, you multiply the degree measure by \(\pi/180^\circ\).
For instance, to convert the given angle \(\pi / 6\) radians from the exercise into degrees, we use the conversion factor: \((\pi / 6) \times (180^\circ/\pi) = 30^\circ\). Being able to convert angles helps in plotting points in the polar coordinate system since protractors typically measure angles in degrees, whereas mathematical functions often use radians.

Graphing in Polar Coordinates

Graphing in polar coordinates involves plotting points based on an angle and a radius from the origin. It's essential when dealing with problems involving periodic functions, circular motion, or any context where the relationship between angles and distances is more intuitive than Cartesian coordinates.
When plotting a point such as \(3, \pi / 6\), you need to graphically represent the angle and the radial distance. First, measure the angle from the positive x-axis. In this case, \(\pi / 6\) radians is converted to 30 degrees using the method from the previous section. Then, mark the angle on your polar grid. After establishing the angle, measure outwards from the pole along the line of the angle to the given radial distance, which is 3 units in this example. Where this distance along the angle line intersects with the circle of radius 3 is the location of the point.
To aid in graphing, polar graphs often have concentric circles indicating different radial distances and lines emanating from the pole to represent different angles. This system is especially beneficial in representing complex waveforms, satellite orbits, or even simple circular paths.