Problem 130 - Calculus Volume 3

Question

Plot the point whose polar coordinates are given by first constructing the angle \(\theta\) and then marking off the distance \(r\) along the ray. $$ \left(2, \frac{5 \pi}{6}\right) $$

Step-by-Step Solution

Answer

Plot the point 2 units at angle \(\frac{5 \pi}{6}\) from the origin.

Step 1: Understanding Polar Coordinates

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

Step 2: Identify the Given Values

Extract the values from the polar coordinates \((2, \frac{5 \pi}{6})\). Here, \(r = 2\) and \(\theta = \frac{5 \pi}{6}\).

Step 3: Construct the Angle

Measure the angle \(\theta = \frac{5 \pi}{6}\) from the positive x-axis. This angle is less than \(\pi\), indicating it lies in the second quadrant. In degrees, \(\frac{5 \pi}{6}\) converts to 150 degrees (since \(\pi\) radians equals 180 degrees).

Step 4: Mark Off the Distance

Starting from the origin, mark a distance of \(r = 2\) units along the ray that extends through the angle \(\frac{5 \pi}{6}\). This is typically done by measuring from the origin outwards along the line at the given angle.

Step 5: Plot the Point

The point is plotted at 2 units from the origin along the ray specified by the angle \(\theta = \frac{5 \pi}{6}\). Ensure the point is located in the correct quadrant as constructed during the angle measure.

Key Concepts

Angle Measurement in Radians

Radians are a way to measure angles, much like degrees, but more naturally used in mathematics. Unlike degrees, which split a circle into 360 parts, radians consider the radius of the circle. One full circle is equivalent to \(2\pi\) radians, meaning \( \pi \) radians is half the circle, just like 180 degrees.
In the given polar coordinate \( \left(2, \frac{5 \pi}{6}\right) \), the angle \( \theta = \frac{5\pi}{6} \) is expressed in radians. To visualize this, remember that \( \pi \) radians equals 180 degrees. So, \( \frac{5 \pi}{6} \) radians is equivalent to dividing 180 degrees into 6 parts and taking 5, which is 150 degrees.
This makes understanding angles in terms of radians helpful for advanced math work, including calculus. It aligns well with mathematical constants like \( \pi \). For visualization, thinking of the angle in degrees might help when plotting or identifying which quadrant the angle points towards.

Quadrant Identification

The coordinate plane is split into four quadrants, identified by the direction of the x and y axes.
The quadrants are labeled I to IV, moving counterclockwise starting from the positive x-axis:

• Quadrant I: Both x and y values are positive.
• Quadrant II: x values are negative, y values are positive.
• Quadrant III: Both x and y values are negative.
• Quadrant IV: x values are positive, y values are negative.

When given the polar angle \( \theta = \frac{5\pi}{6} \), it is crucial to determine which quadrant this angle lies in. Since the angle \( \frac{5\pi}{6} \) equals 150 degrees, which is more than 90 degrees but less than 180 degrees, the direction lands clearly in Quadrant II. This information aids in correctly plotting the point derived from polar coordinates.

Plotting Points

When plotting a point using polar coordinates, it is similar to locating a point on a map but using a radial distance and angle instead of grid coordinates.
The coordinates \( (r, \theta) \) direct you precisely on where to place your point:

• Identify \( \theta \): The angle determines the direction of the point from the origin.
• Measure the angle from the positive x-axis: This gives the line on which to measure \( r \).
• Move along the line: From the origin, measure the radial distance \( r \) along the ray.

For the given polar coordinates \( (2, \frac{5 \pi}{6}) \), first construct the angle \( \frac{5 \pi}{6} \). Then, from the origin, measure 2 units out along the ray defined by this angle. Ensure the measure lies within Quadrant II based on your angle reading.

Coordinate Conversion

Converting between polar coordinates and Cartesian coordinates (the standard x, y system) is a common task and involves a bit of calculation.
This can be useful when working in different coordinate systems and needing to perform algebraic operations or modify visualizations based on provided data.
To convert polar coordinates \( (r, \theta) \) to Cartesian coordinates (x, y), use the formulas:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

In our example with \( r = 2 \) and \( \theta = \frac{5 \pi}{6} \):

• \( x = 2 \cos \left( \frac{5 \pi}{6} \right) = 2 \times \left(-\frac{\sqrt{3}}{2}\right) = -\sqrt{3} \)
• \( y = 2 \sin \left( \frac{5 \pi}{6} \right) = 2 \times \left(\frac{1}{2}\right) = 1 \)

Thus, the Cartesian coordinates are \( (-\sqrt{3}, 1) \), confirming the point's location in Quadrant II.