Problem 21 - Thomas Calculus

Question

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26 .\) $$0 \leq \theta \leq \pi, \quad r=1$$

Step-by-Step Solution

Answer

The graph is a semicircle with radius 1, situated in the top half of the polar plane.

Step 1: Understanding the Polar Equation

The given polar equation is \( r = 1 \). This means that the radius (distance from the origin) for the points on this graph is always 1.

Step 2: Angle Range Interpretation

The angle \( \theta \) is restricted by the inequality \( 0 \leq \theta \leq \pi \). This means that we only consider angles from 0 radians to \( \pi \) radians. \( \pi \) radians is the same as 180 degrees, covering the top half of the polar coordinate system.

Step 3: Identifying Points for Specific Angles

Points on the polar graph can be identified for specific angles within the range. For example, when \( \theta = 0 \), the point is \((1, 0)\) in rectangular coordinates. Similarly, at \( \theta = \pi \), the point is \((-1, 0)\). The points \( (r = 1, \theta) \) form a semi-circle.

Step 4: Plotting the Points on Polar Coordinate System

The polar graph for \( r = 1 \) and \( 0 \leq \theta \leq \pi \) is a half-circle or semicircle centered at the origin with a radius of 1, lying on the top half of the polar coordinate system.

Step 5: Graph Verification

To ensure the plot is correct, check that all points \((1, \theta)\) for \( 0 \leq \theta \leq \pi \) appear on the line, forming a semicircle. Reconfirm the radius remains constant at 1.

Key Concepts

Graphing Polar Equations

Polar coordinates are a way of representing points on a plane using a distance from a starting point (the origin) and an angle from a starting direction. The equation given in our exercise is simple and elegant: \( r = 1 \). This equation tells us that all points on the graph will be exactly 1 unit away from the origin. This distance is consistent for every angle we choose, from \( \theta = 0 \) to \( \theta = \pi \).

To visualize this, picture the origin as the center of a circle. No matter which direction you point to, the distance of 1 unit stays the same. In polar coordinate terms, you are essentially drawing a circle, but in this specific exercise, only half of it is visible due to the angle restriction. Thus, interpreting and drawing polar equations becomes more intuitive once you understand the behavior of \( r \) and \( \theta \). Polar graphing is a step towards more comprehensible and dynamic graph plotting.

Angles in Polar Coordinates

In polar coordinates, angles are just as important as distances. Angles in this system are measured in radians or degrees, with one full rotation being equal to \( 2\pi \) radians or 360 degrees. The angles dictate the direction of the radial line from the origin, providing a path along which \( r \) is measured.

For our exercise, the angle \( \theta \) is set between 0 and \( \pi \). Denoted in radians, this range corresponds to directions from the positive x-axis to the negative x-axis on a Cartesian plane. In polar terms, you're covering all angles in the top half of the graph. It's crucial to understand how these angles translate between different coordinate systems for precision in graphing and plotting. Recognizing the importance of this angle range enables you to grasp how the semicircle in our exercise is formed, since it limits our plotting only to these directions.

Semicircle in Polar Graphs

In the context of our exercise, when plotting the points with \( r = 1 \) and \( 0 \leq \theta \leq \pi \), we form a distinctive shape: a semicircle. A typical circle drawn using polar coordinates \( r = 1 \) is a complete loop. However, the restriction in angle to \( 0 \leq \theta \leq \pi \) results in a plot that covers only the same area's top half, hence forming a semicircle.

This semicircle is centered at the origin with a radius of 1. The shape lies entirely on the positive side of the horizontal axis in the polar plane. By analyzing each point that satisfies \( r = 1 \) within the given angle range, you can effectively map out this half-circle. Each point forms part of this semicircular path, creating a full visual representation of the relation \( 0 \leq \theta \leq \pi \) and \( r = 1 \). Understanding the geometric implication of polar equations, such as forming circles and semicircles, adds another layer of depth to their conceptual clarity.