Question

Sketch the following sets of points \((r, \theta)\). \(0

Short answer

Question: Sketch a set of points in polar coordinates where the radial distance (r) ranges from 0 to 3, and the angle (θ) ranges from 0 to π, then describe the resulting shape. Answer: When sketching the given range of polar coordinates, we get a semi-circle of radius 3 centered at the origin on the Cartesian plane. The semi-circle starts at the positive x-axis and ends at the negative x-axis, indicating the angle range of 0 ≤ θ ≤ π.

Step-by-step solution

Understand Polar Coordinates

Polar coordinates are a way of representing points in a two-dimensional plane using a radial distance (r) and an angle (θ) measured counterclockwise from the positive x-axis. The relationship between polar coordinates (r, θ) and Cartesian coordinates (x, y) is given by: x = r*cos(θ) y = r*sin(θ)

Define the Range of Radial Distance and Angle

We are given the range for the radial distance and angle as follows: - Radial distance (r): 0 < r < 3 - Angle (θ): 0 ≤ θ ≤ π

Identify the Shape

Since the given conditions define all points where the radial distance is between 0 and 3 and the angle is between 0 and π, it will result in a semi-circle of radius 3 centered at the origin.

Sketch the Shape on a Cartesian Plane

To sketch this shape on a Cartesian plane, follow these steps: 1. Draw the positive x-axis from the origin to x = 3 (as θ = 0). 2. Draw the negative x-axis from the origin to x = -3 (as θ = π). 3. Draw the arc of the semi-circle of radius 3, starting on the positive x-axis, and following through the second quadrant, ending on the negative x-axis. 4. You will now have a semi-circle of radius 3, centered at the origin, on the Cartesian plane.

Indicate the Range of Radial Distance and Angle on the Graph

Finally, on the graph, use arrow lines or labels to indicate the following: - The range of radial distance on the semi-circle (0 < r < 3) - The range of angle (0 ≤ θ ≤ π) The resulting graph will show the set of polar coordinates \((r, θ)\), where \(0<r<3\) and \(0 \leq θ \leq \pi\).

Key concepts

Sketching Polar Coordinates

Polar coordinates provide a unique method of representing the location of points on a plane, utilizing a combination of radial distance and angle measurements. To sketch polar coordinates, one must first understand that every point is defined by two values: the radial distance, denoted as r, and an angle, denoted as θ. The radial distance is the length of a straight line from the origin (the center point of the graph) to the point, while the angle is measured in radians or degrees from the positive x-axis.

For the exercise, the radial distance is limited to values between 0 and 3, meaning the point cannot be at the origin (since that would correspond to r = 0) and cannot be beyond a certain distance from the origin. Additionally, the angle θ is restricted from 0 to π, signifying that the points will only be found in the upper half of the plane. The region that satisfies both conditions is a segment of a disk, excluding the center, which forms a semi-circle above the x-axis from the origin.

Converting Polar to Cartesian Coordinates

Conversion from polar coordinates (r, θ) to Cartesian coordinates (x, y) enables us to visualize the points on the more familiar Cartesian plane. By using the fundamental trigonometric relationships, you can translate the radial distance and angle into coordinates. The formulas are x = r·cos(θ) and y = r·sin(θ).

To give a practical example, if you have a point with polar coordinates (2, π/4), you would convert to Cartesian coordinates by calculating x = 2·cos(π/4) = 2/√2 and y = 2·sin(π/4) = 2/√2, giving the Cartesian coordinates (2/√2, 2/√2). This is a straightforward process when dealing with angles that correspond to the standard positions on the unit circle.

Radial Distance and Angle Range

The radial distance and angle range are fundamental to defining regions within polar coordinates. The radial distance, as previously mentioned, determines how far from the origin the plot extends. An inequality such as 0 < r < 3 implies that the points must be inside the circle with a radius of 3 units but not including the circle's boundary nor the origin itself.

The angle range specifies the arc of the circle to consider. In the case of 0 ≤ θ ≤ π, all points lying within the upper half of the circle are included, meaning we consider the area from the positive x-axis (θ = 0) counterclockwise to the negative x-axis (θ = π). The combination of these ranges creates a specific shape or region on the polar plane such as a sector, a segment, or an annulus part.

Graphing in Polar Coordinates

When graphing in polar coordinates, points are placed relative to the origin, called the pole, based on the radial distance and the angle formed with the positive x-axis. For a semi-circle with a range of 0 < r < 3 and 0 ≤ θ ≤ π, you would begin by marking a point on the origin. From there, move outward along the positive x-axis (since θ = 0 at this point) until you reach the boundary of 3 units away.

Next, maintaining the radius at 3, you would sweep counterclockwise up to the negative x-axis where θ = π. This sweeping motion should form a semi-circle above the x-axis. This visual representation explains why polar graphs can sometimes more effectively represent certain equations or shapes than Cartesian coordinates. For example, a full circle or a Lemniscate curve (in the shape of an infinity symbol) can be very easily represented with a simple polar equation, whereas their Cartesian counterparts might be more complex.