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Chapter 8: Problem 32

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

Short Answer

Expert verified
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 by step solution

01

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 \(\frac{5\pi}{6}\).
02

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 \(\theta\).
03

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.

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Most popular questions from this chapter

Find the area of the circle given by \(r=\sin \theta+\cos \theta\). Check your result by converting the polar equation to rectangular form, then using the formula for the area of a circle.

Find two different sets of parametric equations for the rectangular equation. $$ y=\frac{2}{x-1} $$

Write a short paragraph describing how the graphs of curves represented by different sets of parametric equations can differ even though eliminating the parameter from each yields the same rectangular equation.

True or False. Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The graph of the parametric equations \(x=t^{2}\) and \(y=t^{2}\) is the line \(y=x\).

In Exercises \(7-16,\) find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. \(r=\frac{6}{1+\cos \theta}\)
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