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Chapter 8: Problem 10

Graph the point on a polar grid. $$\left(1.2,-\frac{2 \pi}{3}\right)$$

Short Answer

Expert verified
Plot the point at 1.2 units from the origin, at an angle of \(-\frac{2\pi}{3}\) radians.

Step by step solution

01

Understand Polar Coordinates

Polar coordinates are given in the form \((r, \theta)\), where \(r\) is the radius, and \(\theta\) is the angle.
02

Identify the Given Values

The given polar coordinate is \((1.2, -\frac{2\pi}{3})\). Here, \(r = 1.2\) and \(\theta = -\frac{2\pi}{3}\).
03

Plot the Radius

On the polar grid, locate the circle corresponding to \(r = 1.2\). This means you are at a distance of 1.2 units from the origin.
04

Determine the Angle

The angle is \(-\frac{2\pi}{3}\). Since it is negative, measure the angle clockwise from the positive x-axis. \(-\frac{2\pi}{3}\) radians is equivalent to \(-120^\text{掳}\).
05

Locate the Point

From the origin, move 1.2 units along the direction of \(-\frac{2\pi}{3}\) radians (or \(-120^\text{掳}\)). Mark the point where these coordinates intersect on the polar grid.

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Key Concepts

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

Polar Grid
Imagine a polar grid as a series of concentric circles centered at the origin (0,0). Unlike a Cartesian grid, which uses squares and rectangles, a polar grid uses circles and angles. Each circular path represents a different radius, and the lines emanating from the center indicate varying angles. This is essential for understanding how we plot points using polar coordinates.
Radius
The radius, denoted as \( r \), is the distance from the origin to the plotted point. In the exercise, the given radius is 1.2. This means you measure 1.2 units away from the center of the grid. The radius is always positive and determines how far out the point is on the polar grid.
Angle Measurement
Understanding angle measurement is crucial for working with polar coordinates. Angles are measured from the positive x-axis (horizontal axis) and can go either clockwise (for negative angles) or counterclockwise (for positive angles). Angles not only tell you the direction to move on the grid but also how many degrees or radians you turn from the base axis.
Radians
Radians are a way to measure angles based on the radius of a circle. One full circle or revolution equals \( 2\text{蟺} \) (approximately 6.28318 radians). Understanding radians is important in higher mathematics and physics. For example, in the exercise, the angle \( -\frac{2\text{蟺}}{3} \) indicates that the point should be plotted in a clockwise direction, avoiding the dependency on degrees.
Negative Angles
Negative angles are measured in the clockwise direction from the positive x-axis. In the exercise, the angle is given as \( -\frac{2\text{蟺}}{3} \), which translates to \( -120^\text{掳} \). This kind of angle measurement helps you when it is essential to measure backward or clockwise around the polar grid from the initial line.

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Most popular questions from this chapter

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