Problem 20 - Precalculus Mathematics for Calculus

Question

Sketch a graph of the polar equation, and express the equation in rectangular coordinates. $$\theta=5 \pi / 6$$

Step-by-Step Solution

Answer

The polar equation \(\theta = \frac{5\pi}{6}\) converts to the rectangular equation \(y = -\frac{\sqrt{3}}{3}x\). It's a line through the origin with a negative slope.

Step 1: Understanding the Polar Equation

The given polar equation is \(\theta = \frac{5\pi}{6}\). This represents an angle in polar coordinates, suggesting a line that extends indefinitely at this angle from the origin.

Step 2: Converting Polar to Rectangular Coordinates

To convert the equation from polar to rectangular coordinates, we use the relation \( \tan(\theta) = \frac{y}{x} \). Substituting \(\theta = \frac{5\pi}{6}\), we find \( \tan\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{3} \). The rectangular equation becomes \( y = -\frac{\sqrt{3}}{3}x \).

Step 3: Sketching the Graph

To sketch the graph based on the rectangular equation \( y = -\frac{\sqrt{3}}{3}x \), note that it is a straight line passing through the origin with a negative slope of \(-\frac{\sqrt{3}}{3}\). Plot the origin and use the slope to determine additional points (e.g., one unit to the right, move down \(\frac{\sqrt{3}}{3}\) units). The line continues in both directions without bounds.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are used to describe the position of a point by specifying how far it is along the x-axis and the y-axis. When we talk about converting polar coordinates to rectangular coordinates, we're essentially moving from a circular grid system to a square one.
Recap the basics:

• The x-coordinate shows the horizontal placement of the point.
• The y-coordinate indicates the vertical placement of the point.
• The relationship between polar and rectangular coordinates can be found using the tangent function, as shown by the equation: \( \tan(\theta) = \frac{y}{x} \).

In our example, the polar angle \( \theta = \frac{5\pi}{6} \) was used with the tangent relationship, resulting in the rectangular equation \( y = -\frac{\sqrt{3}}{3}x \). This equation describes a straight line, which is infinite in both directions, making it easy to extend our understanding to infinite space just like the polar system. By converting to rectangular coordinates, we gain intuitive insight into plotting and analyzing the graph's behavior.

Graph Sketching

Graph sketching is a powerful tool for visualizing equations and understanding their properties. For the equation \( y = -\frac{\sqrt{3}}{3}x \), sketching the graph involves creating a straight line with a defined slope.
Here's a quick guide to sketch this line:

• Begin at the origin \(0,0\). This is the point where the line crosses both the x-axis and y-axis.
• Understand the slope: A slope of \(-\frac{\sqrt{3}}{3}\) means that for every unit you move to the right (along the x-axis), you'll move \(-\frac{\sqrt{3}}{3}\) units down (along the y-axis).
• To sketch, simply use this slope: from the origin, move one unit to the right and \(-\frac{\sqrt{3}}{3}\) units down and mark the point.
• Repeat this process to establish multiple points along the line. Connect these points with a straight line, extending in infinity in both directions.

Because this line extends through the origin and follows the slope derived from the tangent of our original polar angle, the line embodies the visual representation of the polar equation's direction and inclination.

Angle Conversion

Angle conversion is crucial when moving between different types of coordinate systems. In this exercise, we start with a polar angle \( \theta = \frac{5\pi}{6} \). Converting angles to and from radians and degrees helps in understanding their implications in both systems.
Steps for converting angles:

• Remember that \( \pi \) radians is equal to 180 degrees.
• So, \( \frac{5\pi}{6} \) is equivalent to \( 150 \) degrees, giving us a clear picture in degree-based navigation.
• This conversion is useful because it allows one to make sense of the angle in both mathematical equations and real-world contexts.

When the angle in radians is known, converting it to degrees can help visualize or apply the angle through different mediums, like a protractor. The negative tangent value, resulting from the rectangular coordinates conversion, directly ties back to this angle, showcasing how important angle conversion is for maintaining consistency and intuition across various analyses.