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Chapter 6: Problem 98

In Exercises \(91-116\), convert the polar equation to rectangular form. $$\theta=5 \pi / 6$$

Short Answer

Expert verified
The rectangular form of the polar equation \( \theta = 5\pi / 6 \) is \( x < 0 \) .

Step by step solution

01

Understand the given polar coordinate

\(\theta = 5\pi / 6\) is eactly as it looks - a specific angle. This is not a curve but a straight line from the origin making an angle of \(5\pi / 6\) with the positive x-axis.
02

Convert the Polar equation to Rectangular form

Since there is no radius \(r\) in the equation and it's a standard angle, the rectangular equation will represent a vertical or horizontal line. This is because all points at angle \(\theta\) will lie on this line, regardless of the radius \(r\). In this case, the angle \(5\pi / 6\) is \(150\) degrees, which is in the second quadrant. Here, \( x = r cos \theta \) , since \( \theta \) is in the second quadrant, \(x\) must be negative. Therefore, the rectangular equivalent is \(x < 0\) .

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Coordinates
Polar coordinates offer an alternative way of representing points in a plane using a distance and an angle. Instead of describing a location through horizontal and vertical distances, polar coordinates rely on:
  • The radius ( ) - The distance from the origin (0,0) to the point.
  • The angle ( heta) - The direction from the positive x-axis to the point.
Each point in a polar coordinate system is expressed as \(r, heta\). The radius can be any non-negative number, and the angle is typically measured in radians. Polar coordinates come in handy when working with problems involving circular or rotational symmetry.
For example, a point represented by the coordinates \(r = 3, heta = rac{pi}{3}\) tells us that the point is located 3 units from the origin at a 60-degree angle in the standard position.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, describe a point in a plane using two perpendicular lines intersecting at the origin. The system uses:
  • The x-coordinate - The horizontal distance from the y-axis.
  • The y-coordinate - The vertical distance from the x-axis.
A point is expressed in rectangular coordinates as \(x, y\). This method is familiar to many since it aligns with graph paper grids and is used frequently in algebra and geometry.
The advantage of rectangular coordinates is their simplicity in expressing straight lines and shapes that align with the standard grid system. However, they may become cumbersome when dealing with curves or shapes that are best described in a circular manner, where polar coordinates might offer a more intuitive approach.
Angle Conversion
Angle conversion is the process of changing an angle's measurement from one unit to another, commonly from degrees to radians or vice versa. Understanding angle conversion is crucial when working between different coordinate systems:
  • Degrees - A full circle is divided into 360 degrees.
  • Radians - A circle's circumference divided by its radius equals \(2\pi\), so a full circle spans \(2\pi\) radians.
The conversion between radians and degrees can be a critical step when interpreting polar coordinates, as angles in polar form are usually given in radians. The conversion formulas are:
  • From degrees to radians: \(rad = deg \times \frac{\pi}{180}\)
  • From radians to degrees: \(deg = rad \times \frac{180}{\pi}\)
This conversion helps in understanding and transforming polar equations, such as converting \(\theta = 5\pi/6\) radians, which translates to 150 degrees in the context of rectangular coordinates.

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Most popular questions from this chapter

In Exercises \(71-90,\) convert the rectangular equation to polar form. Assume \(a > 0\). $$y=-2$$

Find the distance between the parallel lines. (Graph can't copy) $$\begin{aligned} &x+y=1\\\ &x+y=5 \end{aligned}$$

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In Exercises \(71-90,\) convert the rectangular equation to polar form. Assume \(a > 0\). $$y=1$$
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