Problem 111 - Precalculus with Limits

Question

In Exercises 109-118, describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. \(\theta=\pi/6\)

Step-by-Step Solution

Answer

The graph of the polar equation \(\theta = \pi/6\) is a line passing through the origin making an angle \( \pi/6\) with the positive x-axis. Its rectangular equation is \(y = x(\sqrt{3}/3)\).

Step 1: Understanding the polar equation

This polar equation is quite simple, the only value given in the equation is \(\theta = \pi/6\). The angle \(\theta\) is set as a constant \(\pi/6\), which means that the radius \(r\) can have any value and the counterclockwise open angle from the positive x-axis is always \(\pi/6\). This represents a line passing through the origin, creating an angle of \(\pi/6\) with the positive x-axis.

Step 2: Convert to rectangular coordinates

In rectangular coordinates, \(\theta = \pi/6\) can be represented as \(y = x \tan(\pi/6)\). The constant angle from the x-axis leads us to express y as x multiplied by the tangent of the constant angle. As \(\tan(\pi/6) = \sqrt{3}/3\), this gives us the equation \(y = x (\sqrt{3}/3)\).

Step 3: Sketching the graph

The final step is to sketch the graph of \(y=x(\sqrt{3}/3)\). This line has a slope of \(\sqrt{3}/3\), which is positive, therefore the line will be an upward sloping line crossing the origin. The line will make an angle of \(\pi/6\) radians (or 30 degrees) with the x-axis as this is the constant angle in the polar coordinate.

Key Concepts

Rectangular Coordinates

Rectangular coordinates are an essential part of our everyday understanding of graphs and geometry. In this form, every point is described by two numbers: \(x\) (the horizontal value) and \(y\) (the vertical value). Together, they form a coordinate \((x, y)\) on the Cartesian plane.

The exercise begins with a polar equation, which is characterized by the angle \(\theta = \pi/6\). This angle signifies the direction in the polar coordinate system. To convert this to a rectangular equation, we need to express the polar description in terms of \(x\) and \(y\). Using the trigonometric identity that \(y = x \tan(\theta)\), and knowing that \(\tan(\pi/6) = \sqrt{3}/3\), we arrive at the rectangular equation for a line: \(y = x (\sqrt{3}/3)\).

This linear equation represents the same line in a way that's more familiar in algebraic contexts. So, when you switch from polar to rectangular coordinates, you are essentially translating a circular method of description to a linear one. This allows for easier graphing and interpretation on the Cartesian plane.

Graph Conversion

Graph conversion can often seem intimidating, but it really is about translating one form of information into another, more functional form. Here, it means going from polar to rectangular equations.

When you deal with polar coordinates, you are essentially working on a circular system where an angle \(\theta\) and a radius \(r\) define the position of a point. However, when converting to rectangular coordinates, your focus shifts to lines on a flat plane using \(x\) and \(y\).

To convert the given polar equation \(\theta = \pi/6\) to a rectangular one, it implies finding a relationship between \(x\) and \(y\) that stays true to the angle of \(\theta\). The expression \(y = x \tan(\pi/6)\) efficiently does that. Recognizing that \(\tan(\pi/6)\) simplifies to \(\sqrt{3}/3\) completes the transformation, resulting in the equation \(y = x (\sqrt{3}/3)\).

The conversion highlights the consistent geometry between two types of coordinate systems, allowing the polar concept of angles to be seen as a slope in rectangular terms.

Slope Calculation

Calculating the slope is crucial for understanding how steep a line is on a graph. The slope, often represented as \(m\), indicates how much \(y\) changes for a unit change in \(x\).

In our context, the slope is derived from the polar equation \(\theta = \pi/6\), which translates to the rectangular equation \(y = x (\sqrt{3}/3)\). Hence, the slope \(m\) of the line is \(\sqrt{3}/3\).

• A positive slope means the line ascends as it moves from left to right.
• A slope of \(\sqrt{3}/3\) reflects a moderate incline, specifically an angle of \(30^\circ\) with respect to the x-axis.

Understanding slope is valuable as it directly connects the angle \(\theta\) from polar coordinates to how steep or flat a line appears in the rectangular system of coordinates. This seamless transition helps immensely in sketching and analyzing mathematical graphs efficiently.