Problem 43 - Algebra and Trigonometry

Question

Identify and graph each polar equation. $$ r=2+\sin \theta $$

Step-by-Step Solution

Answer

The polar equation \( r = 2 + \sin \theta \) forms a limaçon without an inner loop.

Step 1: Understand the Polar Equation

The given polar equation is \[ r = 2 + \sin \theta \]. This means that the radius r depends on the angle \( \theta \). To graph this equation, you need to understand how \( r \) varies as \( \theta \) ranges from \( 0 \) to \( 2\pi \).

Step 2: Determine Key Points on the Graph

Evaluate the equation at several key angles (\( \theta \)) to determine points on the graph:- When \( \theta = 0 \), \( r = 2 + \sin 0 = 2 \).- When \( \theta = \frac{\pi}{2} \), \( r = 2 + \sin \frac{\pi}{2} = 2 + 1 = 3 \).- When \( \theta = \pi \), \( r = 2 + \sin \pi = 2 \).- When \( \theta = \frac{3\pi}{2} \), \( r = 2 + \sin \frac{3\pi}{2} = 2 - 1 = 1 \).Repeat for additional angles as necessary, such as \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{3\pi}{4} \).

Step 3: Identify Symmetry

Check for symmetries in the graph. This equation will exhibit symmetry about the polar axis. If you substitute \( \theta \) with \( -\theta \), the form of the equation remains unchanged.

Step 4: Plot the Polar Graph

Using the points determined in Step 2, plot these points on polar coordinates. Connect these points smoothly to form the curve. The equation \( r = 2 + \sin \theta \) will form a limaçon without an inner loop. The graph will bulge outward because \( r \) changes between 1 and 3.

Key Concepts

Polar Coordinates

Polar coordinates are an alternative to Cartesian coordinates for representing points on a plane. Instead of using \(x\) and \(y\) values, each point is determined by a radius \(r\) and an angle \(θ\).
The radius \(r\) measures how far the point is from the origin. The angle \(θ\) measures the direction of the point relative to the positive x-axis, usually in radians.
Polar coordinates are particularly useful in scenarios where symmetry and angles are essential, such as in trigonometry and complex numbers.
Always note that multiple sets of \(r\) and \(θ\) values can represent the same point, considering negative and positive values, and adding or subtracting multiple of \(2\text{π}\).

Graphing Polar Equations

Graphing polar equations involves plotting points where the radius \(r\) varies based on the angle \(θ\). In the given equation \[ r = 2 + \text{sin}(θ) \], we need to understand how \(r\) changes as \(θ\) ranges from \(0 \text{ to } 2\text{π}\).
Start by calculating \(r\) for key angles like \(0, \frac{\text{π}}{2}, \text{π}, \frac{3\text{π}}{2}\).
For instance:

• When \(θ = 0\), \(r = 2 + \text{sin}(0) = 2\).
• When \(θ = \frac{\text{π}}{2}\), \(r = 2 + \text{sin}\big(\frac{\text{π}}{2}\big) = 3\).
• When \(θ = \frac{3\text{π}}{2}\), \(r = 2 - 1 = 1\).

Using these points, you can create a plot on the polar coordinate plane. Always connect these points smoothly. This method helps achieve a clear and correct graph.

Trigonometric Functions

Trigonometric functions like \text{sin}(θ) and \text{cos}(θ) play a vital role in polar equations. In our example \[ r= 2 + \text{sin}(θ) \], the \(r\) value changes because of the \( \text{sin}(θ) \) component.
Trigonometric functions are periodic, meaning they repeat values in regular intervals, typically every \[ 2\text{π} \].
For the equation \[ r = 2 + \text{sin}(θ) \], \(\text{sin}(θ)\) varies between -1 and 1. Consequently, \(r\) ranges from 1 to 3:

• At \(θ = 0\), \(\text{sin}(0) = 0\).
• At \(θ = \frac{\text{π}}{2}\), \(\text{sin}\big( \frac{π}{2}\big) = 1\).
• At \(θ = \frac{3π}{2}\), \(\text{sin}\big( \frac{3π}{2}\big) = -1\).

Understanding this behavior is crucial for correctly plotting the equation and recognizing the shape and key features of the polar graph.

Symmetry in Polar Graphs

Symmetry is an essential aspect of polar graphing. It helps identify patterns and simplify the graphing process.
Polar graphs can display different types of symmetry:

• Symmetry about the polar axis (x-axis in Cartesian coordinates).
• Symmetry about the \(θ = \frac{\text{π}}{2}\) line (y-axis in Cartesian coordinates).
• Symmetry about the origin.

To check for symmetry, replace \(θ\) with \(-θ\) or \(\text {π} - θ \), and see if the equation remains unchanged. For \[ r = 2 + \text{sin}(θ) \], substituting \(θ\) with \(-θ\) doesn't change the equation, showing it is symmetric about the polar axis.
Symmetry helps to simplify plotting since you only need to graph part of the equation and mirror it across the axis of symmetry.