Problem 32 - Essential Calculus

Question

Convert the polar equation to rectangular form and sketch its graph. $$ \theta=\frac{5 \pi}{6} $$

Step-by-Step Solution

Answer

The equivalent rectangular equation of \(\theta = \frac{5\pi}{6}\) is a straight line passing through the origin making an angle of \(\frac{5\pi}{6}\) radians with the positive x-axis.

Step 1: Understand the polar coordinate system

In the polar coordinate system, a point is specified by the radius \(r\) and the angle \(\theta\) formed with the positive x-axis. Based on this understanding, when given \(\theta = \frac{5\pi}{6}\), it informs us that the angle from the positive x-axis is constant and equals \(\frac{5\pi}{6}\).

Step 2: Convert to rectangular form

The conversion from polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\) generally involves applying the equations \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). However, as our equation only involves \(\theta\), and it is constant, we do not have an equation to convert. Instead, we note that the result of our conversion will be a line of constant \(\theta\).

Step 3: Sketch the line

The line with \(\theta = \frac{5\pi}{6}\) in the polar coordinate system is a line that includes all points that form an angle of \(\frac{5\pi}{6}\) with the positive x-axis. In other words, we should draw a straight line from the origin making an angle of \(\frac{5\pi}{6}\) radians, or 150 degrees, with the positive x-axis.