Question

Sketch the following sets of points \((r, \theta)\). \(2 \leq r \leq 8\)

Short answer

Question: Sketch and describe the set of points (r, θ) where the radius r lies between 2 and 8, and the angle θ can change freely. Answer: The sketch represents a polar coordinate system with two circles centered at the origin (0,0). The inner circle has a radius of 2, and the outer circle has a radius of 8. The region between these two circles is shaded, showing the set of points (r, θ) where 2 ≤ r ≤ 8. Multiple radial lines emanating from the origin, each with a different theta value, are drawn to indicate that angle θ can change freely. The graph also includes x and y-axis labels and a coordinate system for clarity.

Step-by-step solution

Convert r to Cartesian coordinates

First, we will convert the polar coordinates (r, θ) to Cartesian coordinates (x, y) so we can represent all points on the graph. The formulas to convert polar to Cartesian coordinates are: \[x = r\cos(\theta)\] \[y = r\sin(\theta)\]

Sketch the minimum and maximum radii

Since the radius \(r\) in this problem ranges from 2 to 8, we will need to sketch two circles with the same origin (0, 0) and radii equal to 2 and 8. On the sketch, draw two circles centered at the origin, and label the inner circle with radius 2 and the outer circle with radius 8.

Indicate that angle theta can change freely

Since the angle \(\theta\) can vary freely, we need to indicate this on our sketch. To do this, draw multiple radial lines emanating from the origin, each with a different theta value, showing that there are a variety of angles that can be formed with r values between 2 and 8.

Shade the region between the circles

Now, we will shade the region between the inner and outer circles. This illustrates the set of points \((r, \theta)\) where \(2 \leq r \leq 8\).

Include axis labels

Lastly, include x and y-axis labels and a coordinate system to make the graph easy to understand. The final sketch will show the shaded region that represents the set of points \((r, \theta)\) where \(2 \leq r \leq 8\).

Key concepts

Cartesian Coordinates

Cartesian coordinates are a familiar way to represent the location of a point in space using a pair of numbers. These numbers are known as \(x\) and \(y\) coordinates. They are incredibly useful when dealing with two-dimensional spaces.

Each point in this coordinate system is determined by how far it is from the x-axis (the horizontal line) and the y-axis (the vertical line). The point (0,0) is particularly special because it is known as the origin, the center of the coordinate system.

• **X-coordinate**: Represents the point's horizontal position.
• **Y-coordinate**: Represents the point's vertical position.

When we convert from polar coordinates \( (r, \theta) \) to Cartesian coordinates, we use the formulas: \[ x = r \cos(\theta) \] \[ y = r \sin(\theta) \] where \(x\) and \(y\) are the Cartesian coordinates, \(r\) is the radius, and \(\theta\) is the angle in radians.

Understanding these conversions is essential for graphing polar equations on a plane, as they assist in drawing accurate representations.

Graph Sketching

Sketching graphs can be an enjoyable and insightful way to understand math concepts. Especially when it comes to representing polar coordinates on a plane, sketching helps visualize the range and behavior of the data points.

For our problem, we have to visualize all points between two circles with different radii in polar coordinates.

• **Identify components**: Begin by determining the essential components such as minimum and maximum radii and the origin.
• **Draw circles**: Sketch two circles centered at the origin, the first with a radius of 2 and the second with a radius of 8.
• **Indicate angle variation**: With lines or arrows, show that the angle \(\theta\) can be any value, implying a full rotation around the origin.
• **Shade the region**: The area of interest where the radius is between 2 and 8 should be shaded to signify the solution region.

Adding axis labels and ensuring clarity in your sketch will reinforce the presentation of the points existing between these radii.

Sketches make mathematical concepts tangible.

Angular Variation

In polar coordinates, angular variation is crucial because it signifies the direction from the origin a point lies. \(\theta\) represents this angle and gives the direction of the point in the plane, rotating counterclockwise from the positive x-axis.

Unlike Cartesian coordinates, where changes in \(x\) or \(y\) move up, down, left, or right, changes in \(\theta\) can direct us going in a circular path. This makes polar coordinates quite versatile and practical in circular or rotational contexts.

• **Angle representation**: Radial symmetry means any \(\theta\) results in the same point at given \(r\).
• **Complete circle**: \(\theta\) fluctuates from 0 to \(2\pi\), representing a full rotation around the origin.
• **Freedom of movement**: Any angle value indicates infinite possible ways to reach points in the shaded area between the circles.

Knowing how angular variation influences a point's position helps accurately depict solution regions in polar graphs.

Circle Radius

Understanding the circle radius is fundamental when examining points in polar coordinates. The radius \(r\) refers to the distance from the origin to any point on the circle's circumference.

For this problem, where \(2 \leq r \leq 8\), the radius constraint means that we are examining points located between two concentric circles. This range defines a circular band around the origin.

• **Inner circle**: \(r = 2\), corresponds to the inner boundary of our area of interest.
• **Outer circle**: \(r = 8\), defines the maximum distance and forms the outer boundary.
• **Points between**: The area between these circles includes all points with a radius greater than or equal to 2 and less than or equal to 8.

This idea of bounding space with radii helps isolate specific areas in polar plots and connects with the broader mathematical themes of distance and inclusion within regions.

Ensuring clarity in understanding the circle radius can make interpreting and graphing polar data much easier.