Problem 25 - Algebra and Trigonometry

Question

Plot each point given in polar coordinates. $$ \left(6, \frac{\pi}{6}\right) $$

Step-by-Step Solution

Answer

Plot the point 6 units from the origin at an angle of \(30\degree\).

Step 1: Understand Polar Coordinates

Polar coordinates are given in the form \(r, \theta\), where \(r\) is the radius (or distance from the origin) and \(\theta\) is the angle measured from the positive x-axis.

Step 2: Identify the Given Coordinates

In the given problem, the polar coordinates are \(6, \frac{\pi}{6}\). This means \(r = 6\) and \(\theta = \frac{\pi}{6}\).

Step 3: Locate the Angle

Convert the angle \(\frac{\pi}{6}\) into degrees if needed. \(\frac{\pi}{6} = 30\degree\). This is helpful for visualizing the position on the polar graph.

Step 4: Plot the Point

Draw a line from the origin making an angle of \(30\degree\) with the positive x-axis. Measure out a distance of 6 units along this line. Place a point at this position.

Key Concepts

Plotting Points in Polar Coordinates

Plotting points in polar coordinates might seem different from the Cartesian system, but it's quite straightforward once you understand the basics. Polar coordinates are represented as \(r, \theta\). Here, \r\ is the distance from the origin, and \theta\ is the angle formed with the positive x-axis. To plot the point \(6, \frac{\pi}{6}\), start by drawing a circular grid (polar graph) with evenly spaced circles and radial lines for angles. Next, locate the angle \theta = \frac{\pi}{6}\ on the circular grid. This is equivalent to 30 degrees. Move outward from the center (origin) by 6 units along this angle. Mark your point at this location. This point is where the radial line at 30 degrees intersects the circle with radius 6.

Angle Conversion in Polar Coordinates

Understanding angle conversion is crucial for working with polar coordinates. Angles in polar coordinates are usually given in radians. To convert radians to degrees, remember that \(1 \text{ radian}= \frac{180}{\pi}\) degrees. For instance, the angle \frac{\pi}{6}\ radians can be converted to degrees by multiplying by \frac{180}{\pi}\, giving \frac{\pi}{6} \times \frac{180}{\pi} = 30\textdegree\. This conversion helps in visualizing and plotting the point correctly on the polar graph. Angle conversion becomes useful when working with more complex problems or when you need to interpret polar coordinates in a familiar degree-based system.

Polar Graphing Essentials

Polar graphing involves plotting points using the polar coordinate system. The key elements include the origin (the center point), radial lines that represent degrees/radians, and concentric circles that indicate radius values. To graph a point like \(6, \frac{\text{\textpi}}{6}\), identify the following steps: First, find the angle \frac{\text{\textpi}}{6}\ (30 degrees) from the positive x-axis; then move along that line to a distance equivalent to the radius (6 units in this case). This system is useful for problems involving circular motion or angle-based positions in fields like physics and engineering. Practice converting between radians and degrees and plotting accordingly to master polar graphing.