Problem 34 - Precalculus

Question

Sketch the graph of the polar equation using symmetry, zeros, maximum \(r\) -values, and any other additional points. $$r=4-3 \sin \theta$$

Step-by-Step Solution

Answer

The graph of the polar equation \(r=4-3 \sin \theta\) takes a shape of a heart or cardioid, has no zeros, is symmetrical along the x-axis, and has minimum and maximum r-values as 1 and 7 respectively.

Step 1: Find Zeros

Set \(r = 0\) to find the zeros of the function. From \(0=4-3 \sin \theta\), we get \(\sin \theta = 4/3\), which is impossible as the sine of any angle is at most 1. So, this function has no zeros.

Step 2: Determine Maximum r-values

The maximum value of \(\sin \theta\) is 1 and the least value is -1. Thus, the least value of \(4-3 \sin \theta\) is obtained when \(\sin \theta = 1\), hence \(r_{min}=4-3*1=1\). Maximum value is obtained when \(\sin \theta = -1\), hence \(r_{max}=4-3*(-1)=7\).

Step 3: Use Symmetry

The graph is symmetric about the x-axis because replacing \(\theta\) with -\(\theta\) results in the same equation: \(4-3 \sin(-\theta) = 4-3 \sin \theta\). Therefore, we only need to graph for \(0 \leq \theta \leq \pi\), and then reflect about the x-axis for \(-\pi \leq \theta \leq 0\).

Step 4: Sketch the Graph

After understanding the behavior of the function from previous steps, start at \(\theta = 0\), move in the clockwise direction and draw the graph for each quarter. Start with \(r=4\) at \(\theta=0\) (directly right), \(\theta = \pi/2\), \(r = 1\) (directly above), \(\theta = \pi\), \(r = 4\) (directly left), \(\theta = 3\pi/2\), \(r=1\) (directly below), and end with \(r = 4\) at \(\theta=2\pi\) (directly right). The graph will be a shape of a heart called a cardioid.

Key Concepts

Understanding Polar Equations

Polar equations represent a different way of plotting points in a plane compared to the more common Cartesian coordinate system. In a polar system, each point is defined by a radius and an angle, rather than an x and y coordinate. The radius, denoted by \(r\), is the distance from the pole (or origin), and the angle \(\theta\) is measured from the positive x-axis. This system is quite handy for plotting circles, spirals, and other shapes that have radial symmetry.
The given polar equation \(r = 4 - 3 \sin \theta\) describes a shape based on these principles. It uses the trigonometric function sine to define the relationship between \(r\) and \(\theta\). Understanding how the angle \(\theta\) affects \(r\) is key to sketching the graph of this equation.

• Polar graphs rotate around a central point.
• Sine and cosine functions control how "tall" or "wide" the graph becomes.

Knowing this, we can construct polar graphs step by step.

Exploring Symmetry in Polar Graphs

Symmetry is a useful trait in graphs making it easier to predict and draw them. In polar graphs, symmetry can help identify reflected or recurring patterns in the shape. For the equation \(r = 4 - 3 \sin \theta\), it demonstrates symmetry about the x-axis.
To verify this symmetry, we replace \(\theta\) with \(-\theta\). If the equation remains unchanged, the graph is symmetric about the x-axis. For our equation:\[4 - 3 \sin(-\theta) = 4 - 3 \sin \theta\]This confirms that the shape will mirror itself across the x-axis.

• Revolution symmetry: If replacing \(\theta\) with \(-\theta\) yields the same equation.
• Key for efficient graphing: Only half the graph needs to be plotted.
• Reflect across the relevant axis to complete the shape.

Understanding symmetry streamlines the graphing process and reveals the elegant patterns hidden within polar equations.

Finding Maximum r-values

Finding the maximum and minimum values of \(r\) in a polar graph is crucial for understanding its overall shape and size. The maximum \(r\) value for the equation \(r = 4 - 3 \sin \theta\) is derived from the values that \(\sin \theta\) can take, specifically, ranging between -1 and 1.
At \(\sin \theta = -1\), \(r\) reaches its maximum value:\[r_{\text{max}} = 4 - 3(-1) = 7\]And at \(\sin \theta = 1\), \(r\) reaches its minimum value:\[r_{\text{min}} = 4 - 3(1) = 1\]

• Maximum and minimum \(r\) values highlight height and depth of the graph.
• Determine the outermost points and how far they reach from the pole.
• Essential to plot these points for accurate graph representation.

These extremes guide the drawing and understanding of polar graphs and show how functions like sine affect radial distances.

Recognizing Cardioid Shapes

A cardioid is a specific type of polar graph that resembles a heart shape. It is named cardioid due to its close resemblance to the shape of a heart. Cardioids are typically expressed in equations like \(r = c(1 \pm \sin \theta)\) or \(r = c(1 \pm \cos \theta)\). The equation \(r = 4 - 3 \sin \theta\) is a variation of this form.
Recognizing the shape is the first step in understanding and sketching a cardioid. For this exercise, after plotting key points obtained from maximum and minimum \(r\), and confirming symmetry, this equation forms the recognizable heart-like pattern.

• Starts with knowing formula structure: typically includes \(1 \pm \sin\) or \(1 \pm \cos\).
• Cardioids have a single cusp, or point, extending outwards.
• Total rotation around the pole helps complete its unique curve.

The beauty of a cardioid in polar graphs lies in its symmetry and simplicity, demonstrating varied curve behaviors through basic trigonometric functions.