Problem 35 - Precalculus

Question

Sketch the graph of the polar equation using symmetry, zeros, maximum \(r\) -values, and any other additional points. $$r=1-2 \sin \theta$$

Step-by-Step Solution

Answer

The drawn graph should represent a limaçon. The zeroes of the graph fall at θ values of π/6 and 5π/6. The maximum r-value is 3 and occurs at θ=π, and observing the symmetry with respect to the x-axis, we can sketch the complete graph of the given polar equation.

Step 1: Understanding the Polar Equation

Recognize that the polar equation \(r=1-2 \sin \theta\) is not in standard form. The standard forms for polar equations that produce certain types of graphs are \(r=a \pm \sin \theta\), or \(r=a \pm \cos \theta\). So, this function will produce a graph known as a 'limaçon'.

Step 2: Identifying the Zeros

The zeros of the equation are found when \(r = 0\). Hence, solve \(1-2\sin\theta = 0\) for \(\theta\). Hence, we find that \(\sin\theta = 1/2\). So, \(\theta = \pi/6\) or \(\theta = 5\pi/6\).

Step 3: Determining the Maximum r-values

Since the equation is in the form of \(r = a - b \sin \theta\), the maximum of \(r\) is \(a + b = 1 + 2 = 3\) and the minimum of \(r\) is \(a - b = 1 - 2 = -1\). However, since \(r\) describes a distance, and is thereby always non-negative, we can rule out \(r = -1\).

Step 4: Sketching the Graph

Make an \(r-\theta\) table of values and plot these on the polar grid. Start with the known values from Step 2 and 3, and add a few more to get a clear picture of the graph. Plot these polar coordinates (r, θ) and sketch the curve through these points. Because of the sin term in the polar equation, there is symmetry about the x-axis.

Key Concepts

Graph Sketching

Graph sketching involves creating a visual representation of a mathematical equation in a coordinate system. In polar coordinates, this means plotting points given by their radius \( r \) and angle \( \theta \). The equation \( r = 1 - 2 \sin \theta \) gives us a specific relationship between \( r \) and \( \theta \). To sketch the graph, we create a table of values for \( r \) and \( \theta \), using specific angles to calculate corresponding \( r \) values. This helps to ensure key characteristics of the curve are displayed.
Start by obtaining necessary values like zeros and maximum \( r \)-values by substituting angles into the equation. Gather additional points for more clarity. Once you have your points, plot them on the polar grid. Connect these points smoothly, maintaining the curve's continuous shape. This process paints a detailed picture of the graph's behavior.

Symmetry in Polar Coordinates

Symmetry in polar coordinates simplifies graph sketching by reducing the required calculations. Symmetry helps identify regions where the graph's shape or structure is mirrored. For the equation \( r = 1 - 2 \sin \theta \), check for symmetry about the line \( \theta = \frac{\pi}{2} \), polar axis, and origin.
To quickly determine symmetry, replace \( \theta \) with \( -\theta \), \( \pi-\theta \), or add \( \pi \) to \( \theta \). If the resulting expressions still match the original equation, symmetry exists. In this case, the \( -\sin \theta \) term indicates symmetry across the x-axis as it negates the angle. This greatly aids in graph sketching by revealing mirrored properties and confirming the graph's consistency across the x-axis.

Limaçon Curve

A limaçon curve is characterized by its distinctive "snail" shape. It is defined by equations of the form \( r = a \pm b \sin \theta \) or \( r = a \pm b \cos \theta \). The equation \( r = 1 - 2 \sin \theta \) specifically represents a limaçon with an inner loop because the coefficient of the sine function \( b \) is greater than \( a \).
The limaçon features an inner loop, a dimple, or a convex shape, determined by the relative sizes of \( a \) and \( b \). In our equation, since \( |b| > |a| \), an inner loop forms. Understanding this helps in predicting the curve's behavior and confirming the sketch. Thus, while graphing, expect the curve to loop inwards, cross itself, and extend outward forming this unique shape.

Zeros of Polar Equations

Zeros of polar equations occur where the radial component \( r \) becomes zero. These are crucial because they indicate points where the curve intersects the pole (origin). To find zeros in \( r = 1 - 2 \sin \theta \), set \( r = 0 \) and solve for \( \theta \).
In our example, solving \( 1 - 2 \sin \theta = 0 \) gives \( \sin \theta = \frac{1}{2} \). Therefore, \( \theta = \frac{\pi}{6} \) and \( \theta = \frac{5\pi}{6} \) are the angles where \( r \) is zero. These angles correspond to the zeros that lie directly on the x-axis of the polar grid, crucial for accurately sketching the graph's crossover points. Knowing where zeros occur helps map the graph's significant features and confirm points where the curve returns to the origin.