Problem 20 - Algebra and Trigonometry

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 \(\sqrt{3}x + 3y = 0\).

Step 1: Understanding Polar Coordinates

The given polar equation is \[ \theta = \frac{5\pi}{6}. \]This equation represents a line in the polar coordinate system where each point on that line has the same angle \(\theta\) with the positive x-axis.

Step 2: Converting Polar Equation to Rectangular Coordinates

In polar coordinates, a point \((r, \theta)\) can be expressed in rectangular coordinates \((x, y)\) using the formulas:\[x = r \cos \theta, y = r \sin \theta.\]To find the rectangular equation, use the tangent relationship since \(\theta\) gives the angle:\(\tan \theta = \frac{y}{x} \).Substituting \(\theta = \frac{5\pi}{6}\) gives:\[\tan \left( \frac{5\pi}{6} \right) = \frac{y}{x} \]. Recalling that \(\tan \left( \frac{5\pi}{6} \right) = -\sqrt{3}/3\), we have:\[\frac{y}{x} = -\frac{\sqrt{3}}{3} \].Solving for \(y\), the rectangular equation is:\[ -\sqrt{3}x = 3y \quad \Rightarrow \quad \sqrt{3}x + 3y = 0. \]

Step 3: Sketching the Polar Graph

The graph of the polar equation \(\theta = \frac{5\pi}{6}\) is a straight line that passes through the origin at an angle of \(\frac{5\pi}{6}\) radians from the positive x-axis. This line extends infinitely in both directions of the angle \(\theta = \frac{5\pi}{6}\).

Step 4: Sketching the Rectangular Graph

The rectangular equation \(\sqrt{3}x + 3y = 0\) represents a line with slope \(-\frac{\sqrt{3}}{3}\). This line will pass through the origin (0,0) and cross the y-axis at the same slope.

Key Concepts

Rectangular Coordinates

In mathematical terms, the rectangular (or Cartesian) coordinate system is used to uniquely identify points on a plane. This is done using two values, traditionally called x and y. The x-axis runs horizontally, while the y-axis runs vertically. Together, they create a grid that allows you to express any location as a coordinate pair (x, y). Each point on this plane corresponds to a unique pair of values, which can also describe its position in more complex relationships or equations.

Understanding rectangular coordinates is crucial for converting other coordinate systems, like polar coordinates, into simpler linear equations that describe lines and shapes on this grid. In essence, it provides a straightforward way to visualize and calculate spatial relationships between different elements.

Polar to Rectangular Conversion

The process of converting polar coordinates to rectangular coordinates involves using trigonometric relationships. Polar coordinates use a radius and an angle, the pair \((r, \theta)\), to describe a point's position relative to the origin, distinct from the linear grid used in rectangular coordinates.
To transform this into a rectangular coordinate system, you will rely on the following fundamental equations:

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

These equations use the polar angle \(\theta\) and the radius \(r\) to calculate the corresponding x and y values in rectangular coordinates. It's a method that bridges the gap between circular and linear displays of information. Converting these coordinates allows you to visualize points differently and solve problems that require straight-line graphing.

Graphing Polar Equations

Graphing polar equations can be quite different compared to graphing in the rectangular system. Polar graphs are designed to be circular, where angles and distances measured from the origin define each point. For example, the equation \(\theta = \frac{5\pi}{6}\) depicts a straight line. This line is oriented at an angle of \(\frac{5\pi}{6}\) radians from the positive x-axis, cutting through the origin.

To sketch such a line, imagine the point where this angle intersects the unit circle, and then extend a straight line both ways through the origin. Each point on this line has the same angle with the x-axis, defined by \(\theta\). This graphical representation is simple in polar terms but needs conversion for working in rectangular terms, as seen from our example's conversion to \(\sqrt{3}x + 3y = 0\). Understanding how to interpret and translate these graphs ensures a more robust grasp of mathematics in both systems.