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Chapter 7: Problem 18

Plot the points whose polar coordinates are given. $$\left(2,-180^{\circ}\right)$$

Short Answer

Expert verified
Plot point 2 units to the left of the origin, on the negative x-axis.

Step by step solution

01

Understanding Polar Coordinates

Polar coordinates are given as (r, 胃), where r is the radius (distance from the origin) and 胃 is the angle. The angle is measured in degrees or radians.
02

Identify the Radius and Angle

In the given polar coordinates (2, -180掳), the radius r is 2, and the angle 胃 is -180掳.
03

Convert the Negative Angle if Necessary

The angle -180掳 is equivalent to +180掳, since adding 360掳 (a full circle) to -180掳 yields 180掳.
04

Plot the Point on the Polar Grid

To plot the point, move 2 units away from the origin and rotate 180掳 counterclockwise from the positive x-axis. This point is plotted directly to the left of the origin, on the negative x-axis.

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Key Concepts

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

polar grid
A polar grid is a circular graph used to plot points in polar coordinates. Unlike a Cartesian grid, which has horizontal and vertical lines, a polar grid consists of concentric circles and radial lines. The concentric circles, centered at the origin, represent different values of the radius (r). The radial lines that spread out from the origin represent different angles (胃).
When plotting on a polar grid, you first move out from the origin according to the radius (r), then rotate according to the angle (胃). Understanding the layout of a polar grid is essential for correctly plotting points in polar coordinates.
angle conversion
Angles in polar coordinates can be measured in degrees or radians. Sometimes, you may encounter negative angles. Negative angles are measured clockwise from the positive x-axis, while positive angles are measured counterclockwise.
To convert a negative angle to a positive angle, simply add 360掳 to it. For instance, an angle of -180掳 becomes +180掳 when you add 360掳. Similarly, to convert a positive angle that is greater than 360掳, you can subtract 360掳.
Knowing how to convert angles is crucial because it ensures that you can properly locate points on the polar grid.
radius and angle
In polar coordinates, each point is defined by a radius (r) and an angle (胃). The radius represents the distance from the origin to the point. The angle represents the direction from the origin to the point relative to the positive x-axis.
For example, in the coordinates (2, -180掳), the radius is 2, and the angle is -180掳. To plot this point:
  • First, start from the origin and move outward 2 units.
  • Then, rotate 180掳 counterclockwise (or equivalent to -180掳 clockwise) to place the point on the polar grid.

This point will be positioned on the negative x-axis because the angle is 180掳 from the positive x-axis.

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Most popular questions from this chapter

In a controlled experiment, the temperature is \(0^{\circ} \mathrm{C}\) at time \(t=0 .\) The temperature is increased to \(10^{\circ} \mathrm{C}\) at time \(t=4\) and then decreased to \(-10^{\circ} \mathrm{C}\) at time \(t=12 .\) The temperature returns to \(0^{\circ} \mathrm{C}\) at time \(t=16 .\) Assuming the temperature on the time interval \([0,16]\) is a sine wave, write the temperature \(y\) as a function of the time \(t\)

Find \(z_{1} z_{2}\) and \(z_{1} / z_{2}\) for each pair of complex numbers, using trigonometric form. Write the answer in the form \(a+b i\). $$z_{1}=3-4 i, z_{2}=-1+3 i$$

Find all solutions to the equation \(4 \sin ^{2}(3 x)-3=0\) in the interval \((0, \pi)\)

Write each vector as a linear combination of the unit vectors i and \(\mathbf{j}\). $$\langle 1,5\rangle$$

Write each vector as a linear combination of the unit vectors i and \(\mathbf{j}\). $$\langle- 1 / 2,0\rangle$$
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