Problem 14 - Thomas Calculus

Question

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(7-22\) . $$ \theta=11 \pi / 4, \quad r \geq-1 $$

Step-by-Step Solution

Answer

The graph is a line through the origin at an angle of \( 135^{\circ} \).

Step 1: Understand the Polar Equation

In polar coordinates, an equation can be given in terms of \( r \) (the distance from the origin) and \( \theta \) (the angle from the positive x-axis). This problem involves the angle \( \theta = \frac{11\pi}{4} \) and the radial distance condition \( r \geq -1 \).

Step 2: Convert Angle to Standard Range

Convert \( \theta = \frac{11\pi}{4} \) to an equivalent angle between \( 0 \) and \( 2\pi \) by subtracting \( 2\pi \): \( \frac{11\pi}{4} - 2\pi = \frac{11\pi}{4} - \frac{8\pi}{4} = \frac{3\pi}{4} \). So, \( \theta = \frac{3\pi}{4} \) is the equivalent angle.

Step 3: Interpret the Radial Inequality

The condition \( r \geq -1 \) means that the set of points includes all radial distances from the origin as long as they are greater than or equal to \(-1\). Since \( r \) cannot physically be negative in polar form, this implies all positive \( r \), including \( r = 0 \), and points in the reverse direction of the angle for negative \( r \).

Step 4: Visualize the Graph

For \( \theta = \frac{3\pi}{4} \), draw a line at \( 135^{\circ} \) which extends infinitely forward (for positive \( r \)) and backward in the opposite direction (for negative \( r \)). This line includes the origin because \( r = 0 \) is part of the solution.

Key Concepts

Angle Conversion

Polar coordinates use angles measured from the positive x-axis, often expressed in radians. When given an angle like \( \theta = \frac{11\pi}{4} \), it can be greater than the standard range \( 0 \leq \theta < 2\pi \). To convert it to this range, one needs to determine its equivalent angle by subtracting multiples of \( 2\pi \). For instance, \( \frac{11\pi}{4} - 2\pi \) simplifies to \( \frac{3\pi}{4} \). This means both angles point in the same direction in a polar plot.
Understanding angle conversion is key to interpreting polar coordinates correctly, as it ensures that reference angles are standard and recognizable for graphing or further calculations.

Radial Inequality

Radial inequalities in polar coordinates determine which points are considered in relation to the origin. In this problem, the condition \( r \geq -1 \) includes all points where the radial distance, \( r \), is non-negative. Although we can't have a negative distance from the origin in practical terms, including negative values helps us consider the reverse direction along a line.

• \( r > 0 \): Points are in the forward direction from the origin.
• \( r = 0 \): Represents the origin itself.
• \( r < 0 \): Points in the opposite direction along the same angle.

This interpretation creates a line extending in both directions from the origin in the graph, making it part of the solution."

Graphing Polar Equations

Graphing polar equations starts with plotting the angle \( \theta \) and drawing lines or curves based on the radial conditions. With \( \theta = \frac{3\pi}{4} \), which corresponds to \( 135^\circ \), you draw a line at this angle extending infinitely.
Due to the radial inequality \( r \geq -1 \), this line doesn't just stop at the origin. Instead, it stretches forwards and backwards, reflecting the condition \( r \geq -1 \). The forward section covers all positive \( r \), and the part extended backward accounts for considered points that reflect negative \( r \). Thus, each point lies along a single line passing through the origin, forming the complete graph of the solution.