Problem 26 - Algebra and Trigonometry

Question

Plot each point given in polar coordinates. $$ \left(5, \frac{5 \pi}{3}\right) $$

Step-by-Step Solution

Answer

Plot the point 5 units from the origin at an angle of \(\frac{5\pi}{3}\) radians.

Step 1: Understand the Polar Coordinates

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

Step 2: Identify the Given Coordinates

The given polar coordinates are \((5, \frac{5 \pi}{3})\). This means that \(r = 5\) and \(\theta = \frac{5\pi}{3}\).

Step 3: Convert Angle to Degrees (Optional)

The angle \(\frac{5\pi}{3}\) can be converted to degrees if needed: \ \frac{5\pi}{3} \times \frac{180}{\pi} = 300^{\circ}\.

Step 4: Plot the Point

To plot the point, start at the origin (0,0). Move 5 units away from the origin in the direction of the angle \(\frac{5\pi}{3}\) or \ 300^{\circ} \ (located in the fourth quadrant of the polar coordinate system).

Key Concepts

radians to degrees conversion

Polar coordinates often use radians to measure angles, but it's helpful to understand how to convert radians to degrees. This is especially useful if you are more comfortable working with degrees.
Remember that one full circle is equal to 360 degrees, which is also equal to \(2\pi\) radians. The conversion between radians and degrees can be done using the formula: \[ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \]
To convert an angle from radians to degrees, you multiply by \( \frac{180}{\pi} \). For example, let's convert \( \frac{5\pi}{3} \) radians to degrees:
\[ \frac{5\pi}{3} \times \frac{180}{\pi} = 300^{\circ} \]
This tells us that \( \frac{5\pi}{3} \) radians is equivalent to 300 degrees.

plotting polar points

Plotting a point in the polar coordinate system might seem tricky at first, but with a few steps, it becomes straightforward. Polar coordinates are in the form \( (r, \theta ) \). The first value, \( r \), represents the distance from the origin (0, 0), and the second value, \( \theta \), represents the angle measured from the positive x-axis.
To plot the point \( (5, \frac{5\pi}{3}) \):

• Start at the origin, which is the point (0, 0).
• Move 5 units away from the origin in the direction specified by the angle \( \frac{5\pi}{3} \) or 300 degrees. This angle falls in the fourth quadrant of the polar coordinate system.

By following these steps, you will have accurately plotted the polar point.

angular coordinate

The angular coordinate in polar coordinates is denoted as \( \theta \), and it represents the angle formed with the positive x-axis. This angle can be measured in either degrees or radians.
For instance, in the polar coordinates \( \frac{5\pi}{3} \), \( \theta \) is \( \frac{5\pi}{3} \) radians, which we converted to 300 degrees.
Understanding the angular coordinate helps determine in which quadrant of the polar coordinate system the point lies:

• 0 to 90 degrees (0 to \( \frac{\pi}{2} \)) is the first quadrant.
• 90 to 180 degrees ( \( \frac{\pi}{2} \) to \( \pi \)) is the second quadrant.
• 180 to 270 degrees ( \( \pi \) to \( \frac{3\pi}{2} \)) is the third quadrant.
• 270 to 360 degrees ( \( \frac{3\pi}{2} \) to \( 2\pi \)) is the fourth quadrant.

In our example, 300 degrees places the point in the fourth quadrant.