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Chapter 1: Problem 129

In the following exercises, plot the point whose polar coordinates are given by first constructing the angle \(\theta\) and then marking off the distance \(r\) along the ray. $$\left(1, \frac{\pi}{4}\right)$$

Short Answer

Expert verified
Plot the point 1 unit from the origin at a 45-degree angle to the x-axis.

Step by step solution

01

Understand Polar Coordinates

Polar coordinates are given as a pair \(r, \theta\), where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis. The given coordinates are \(\left(1, \frac{\pi}{4}\right)\), meaning the point is 1 unit from the origin and forms an angle of \(\frac{\pi}{4}\) (or 45 degrees) with the positive x-axis.
02

Construct the Angle \(\theta\)

Start from the positive x-axis, which can be considered as the direction of zero degrees or zero radians. Rotate counterclockwise to form the angle \(\frac{\pi}{4}\), equivalent to 45 degrees. Visualize or use a protractor to find this angle on your coordinate plane.
03

Mark the Distance \(r\)

From the origin (0,0), move along the line that makes a 45-degree angle from the positive x-axis. Measure and mark a point that is 1 unit away from the origin along this direction. This distance \(r = 1\) is straightforward and can usually be marked with a compass or a ruler.
04

Plot the Point

The point is now established by the intersection of the distance \(r=1\) along the angle \(\theta = \frac{\pi}{4}\). Ensure the point is placed accurately at 1 unit from the origin along the line making a 45-degree angle.

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Key Concepts

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

Angle Theta
In polar coordinates, angle \( \theta \) is a fundamental element that helps us determine the direction of a point from the origin.To visualize this, picture the coordinate plane with the central point known as the origin, and imagine a line or ray extending to the right as the positive x-axis.This position represents \(0\) degrees or \(0\) radians. To find angle \( \theta \), start from this positive x-axis and rotate counterclockwise.In our example, \( \theta = \frac{\pi}{4} \), which corresponds to a 45-degree rotation.
  • Counterclockwise movement is considered positive.
  • \( \theta \) defines the angle between the positive x-axis and the line connecting the origin to the point.
  • For angles larger than \(2\pi\), continue rotating past \(360\) degrees accordingly.
Understanding and constructing \( \theta \) is essential before moving on to the next element: distance \( r \).
Distance R
Distance \( r \) in polar coordinates indicates how far the point is from the origin.It's similar to the concept of radius in a circle.For the coordinates \((1, \frac{\pi}{4})\), the distance \( r = 1 \) tells us that the point lies exactly one unit away from the origin, in the direction specified by \( \theta \).
  • \( r \) is always non-negative; a negative \( r \) often means moving in the opposite direction of \( \theta \).
  • A larger \( r \) means the point is farther from the origin; the plot becomes larger in the same angle direction.
  • Use a compass or a ruler for accuracy when plotting distance \( r \).
This distance is crucial in determining the precise location of the point within the coordinate plane.
Coordinate Plane
The coordinate plane is a two-dimensional space where points are plotted based on two reference axes: the x and the y.In polar coordinates, however, we rely on \( r \) and \( \theta \) to plot points instead of traditional x and y coordinates.
  • The origin is the central point, denoted by (0,0) in Cartesian coordinates.
  • For polar plotting, the positive x-axis is your starting reference line.
  • Think of each point as sitting on a ray originating from the origin, defined by \( \theta \) and extending for distance \( r \).
Your understanding of the coordinate plane is enhanced by seeing it as a grid where any point can be precisely identified with these angles and distances.
Plotting Points
Plotting a point in polar coordinates involves combining both the angle \( \theta \) and the distance \( r \).Visualize it as drawing a line from the origin, rotating it to the correct angle, and then marking off the distance.
  • Start by finding \( \theta \) 鈥� rotate your line counterclockwise from the positive x-axis.
  • Next, mark the distance \( r \) from the origin to locate exactly where the point lies on this ray.
  • Check stability and precision; ensure no overlap with other plotted points.
Once you understand this, you'll find plotting points in polar coordinates quite intuitive and similar to marking positions based on directions and distances in real life.

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Most popular questions from this chapter

Find the surface area obtained by rotating \(x=3 t^{2}, y=2 t^{3}, 0 \leq t \leq 5\) about the \(y\) -axis.

Use a CAS to find the area of the surface generated by rotating \(x=t+t^{3}, y=t-\frac{1}{t^{2}}, 1 \leq t \leq 2\) about the \(x\) -axis. (Answer to three decimal places.)

For the following exercises, convert the polar equation to rectangular form and sketch its graph. $$r=\cot \theta \csc \theta$$

For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation. $$r=\csc \theta$$

For the following exercises, sketch a graph of the polar equation and identify any symmetry. $$r=3 \cos \left(\frac{\theta}{2}\right)$$
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