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Chapter 8: Problem 6

Plot the point with the given polar coordinates. $$ \left(\frac{2}{3}, 7 \pi / 4\right) $$

Short Answer

Expert verified
Plot the point \(\frac{2}{3}\) units from the origin at an angle of 315 degrees (45 degrees below the x-axis) in the fourth quadrant.

Step by step solution

01

Understand Polar Coordinates

Polar coordinates are given in the form \((r, \theta)\), where \(r\) represents the radius or distance from the origin, and \(\theta\) represents the angle measured counterclockwise from the positive x-axis. In this case, \(r = \frac{2}{3}\) and \(\theta = \frac{7\pi}{4}\).
02

Convert the Angle to Degrees

The angle given is \(\frac{7\pi}{4}\). To understand this better, convert it to degrees by using the conversion \(1 \text{ radian} = 180/\pi \text{ degrees}\). Thus, \(\frac{7\pi}{4} = \frac{7\times180}{4} = 315 \text{ degrees}\).
03

Visualize the Angle on the Coordinate Plane

315 degrees is equivalent to \(360 - 45\), which means it points in the fourth quadrant in the standard Cartesian plane. This angle is 45 degrees below the positive x-axis.
04

Plot the Point Using the Radius and Angle

From the origin, measure \(\frac{2}{3}\) units along the direction of the angle measured in Step 3. This point will be in the fourth quadrant, closer to the origin compared to points with radius greater than 1.

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Key Concepts

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

Angle Conversion
In polar coordinates, understanding and working with angles is a key aspect. The angle \(\theta\) in a polar coordinate refers to its rotation from the positive x-axis. Often given in radians, it's essential to convert it into degrees to better visualize and interpret. This is because many learners are more familiar with the degree system.
To convert radians to degrees, use the formula:
  • 1 radian = 180/\(\pi\) degrees.
For instance, the angle given as \(\frac{7\pi}{4}\) radians needs to be converted to degrees. Calculating this, we perform the following:
  • \(\frac{7\times180}{4}\) gives 315 degrees.
This conversion establishes a clearer understanding of the angle's direction, especially when identifying its quadrant in the coordinate system.
Plotting Points
Plotting points in the polar coordinate system may seem quite different compared to the Cartesian system, but it involves a few simple steps. In polar coordinates, a point is represented as \( (r, \theta) \), where \( r \) is the radius 鈥� or distance from the origin 鈥� and \( \theta \) is the angle.
To plot a point given in polar coordinates:
  • Locate the angle \(\theta\) on the polar plane first. Start from the positive x-axis, moving counterclockwise.
  • Measure the radius \(r\) from the origin in the direction of the angle found.
In our example, with a radius of \( \frac{2}{3} \) units and an angle of 315 degrees, begin at the origin and move \( \frac{2}{3} \) units in the direction that aligns with 315 degrees, which we have identified lies in the fourth quadrant.
Remember that the radius here is less than one, indicating the point is closer to the origin.
Quadrants in Coordinate Plane
The coordinate plane is divided into four quadrants that help in locating angles and points. Each quadrant corresponds to a specific angle range:
  • First Quadrant: 0 to 90 degrees
  • Second Quadrant: 90 to 180 degrees
  • Third Quadrant: 180 to 270 degrees
  • Fourth Quadrant: 270 to 360 degrees
Understanding the quadrant location is crucial for plotting points correctly.
In our example, we converted the angle \(\frac{7\pi}{4}\) to 315 degrees, which places it in the fourth quadrant. This quadrant is known for angles between 270 and 360 degrees, typically positioned below the positive x-axis. Angles here rotate clockwise and span between the x and y-axes beyond 270 degrees.
Knowing which quadrant your point lies in can guide you on how to plot it accurately, ensuring you measure the angle in the right direction from the origin.

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Most popular questions from this chapter

Suppose \(r=f(\theta)\) is a polar equation. Graphically interpret the given property. $$ f(-\theta)=-f(\theta) \text { (odd function) } $$

Find a rectangular equation that has the same graph as the given polar equation. $$ r^{2}=4 \sin 2 \theta $$

Comet Halley (a) The eccentricity of the elliptical orbit of Comet Halley is 0.97 and the length of the major axis of its orbit is \(3.34 \times 10^{9}\) mi. Find a polar equation of its orbit of the form \(r\) \(=e p /(1-e \cos \theta)\) (b) Use the equation in part (a) to obtain \(r_{p}\) and \(r_{a}\) for the orbit of Comet Halley.

Find the rectangular coordinates for each point with the given polar coordinates. $$ (4, \pi / 8) $$

Find polar coordinates that satisfy (a) \(r>0,-\pi<\theta \leq \pi\) (b) \(r<0,-\pi<\theta \leq \pi\) for each point with the given rectangular coordinates. $$ (\sqrt{6}, \sqrt{2}) $$
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