Problem 26 - Algebra and Trigonometry

Question

Sketch a graph of the polar equation. $$ r=1+\sin \theta $$

Step-by-Step Solution

Answer

The graph of \(r = 1 + \sin \theta\) is a cardioid.

Step 1: Understanding the Polar Equation

The given polar equation is \(r = 1 + \sin \theta \). This means the radius \(r\) depends on the angle \(\theta\), and it varies based on the value of \(\sin \theta \).

Step 2: Analyzing Key Values of \(\theta\)

To understand how \(r\) changes, observe key angles where \(\sin \theta\) takes on characteristic values: \(\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2},\) and \(2\pi\). For \(\theta = 0\), \(\sin 0 = 0\) so \(r = 1\). For \(\theta = \frac{\pi}{2}\), \(\sin \frac{\pi}{2} = 1\), so \(r = 2\). For \(\theta = \pi\), \(\sin \pi = 0\), \(r = 1\). For \(\theta = \frac{3\pi}{2}\), \(\sin \frac{3\pi}{2} = -1\), \(r = 0\). Finally, for \(\theta = 2\pi\), \(\sin 2\pi = 0\), \(r = 1\).

Step 3: Plotting Points in Polar Coordinates

Plot these points on a polar graph: - \((1, 0)\) for \(\theta = 0\)- \((2, \frac{\pi}{2})\)- \((1, \pi)\)- \((0, \frac{3\pi}{2})\)- \((1, 2\pi)\)

Step 4: Drawing the Graph

Connect the plotted points while noting the symmetry. The graph will form a heart-shaped curve known as a cardioid. Starting from \((1, 0)\), it will smoothly increase to \((2, \frac{\pi}{2})\), then decrease back through \((1, \pi)\) and reach \((0, \frac{3\pi}{2})\), and finally loop back to \((1, 2\pi)\).

Step 5: Final Graph Sketch

By following the connected path, you sketch a cardioid shape which has a cusp at the origin \((0, \frac{3\pi}{2})\) and is symmetric with respect to the vertical line through the pole.

Key Concepts

polar equation

A polar equation is a representation of a curve using polar coordinates, which describe the location of a point in a plane using the distance from the origin and the angle from a reference direction (usually the positive x-axis). Unlike Cartesian equations that use (x, y) coordinates, polar equations specify points with (r, \( \theta \)), where \( r \) is the radius or distance from the origin, and \( \theta \) is the angle.

• In the equation \( r = 1 + \sin \theta \), \( r \) determines how far the point is from the origin.
• \( \theta \) dictates the direction of that distance.
• \( \sin \theta \) impacts the length, causing a variation as \( \theta \) changes.

Understanding polar equations involves recognizing how changes in \( \theta \) affect \( r \), which leads to the path of the curve. This equation's simplicity stems from having just one trigonometric component, which typically forms familiar shapes in polar graphs.

cardioid

The term cardioid comes from the Latin word "cardi," meaning heart, due to its heart-like shape. In polar equations, a cardioid is a type of singular point curve which often features a single cusp, or a pointed tip, and symmetry around one axis.
In our exercise with the polar equation \( r = 1 + \sin \theta \), we derive a cardioid by plotting key points and sketching the connecting curve. This graph is distinctive:

• Starts at radius 1 when \( \theta = 0 \).
• Expands to its maximum at radius 2 when \( \theta = \frac{\pi}{2} \).
• Reverse back to radius 1 as \( \theta \) reaches \( \pi \).
• Hits a cusp, or a sharp point, at the origin when \( \theta = \frac{3\pi}{2} \).
• Makes a full loop, ending again at radius 1 when \( \theta = 2\pi \).

This cardioid is symmetric along the vertical axis where \( \theta = \frac{\pi}{2} \), confirming the typical properties of these unique curves. Understanding cardioids allow learners to appreciate symmetry and how trigonometric functions influence curve shapes.

graphing in polar coordinates

Graphing in polar coordinates involves plotting points based on their radial distance and angular direction, rather than horizontal and vertical distances like Cartesian graphs. This method can elegantly display curves like circles, spirals, and cardiods.
To effectively graph the equation \( r = 1 + \sin \theta \):

• Identify key values of \( \theta \) that offer clear landmarks, such as 0, \( \frac{\pi}{2} \), \( \pi \), \( \frac{3\pi}{2} \), and \( 2\pi \).
• Calculate the corresponding \( r \) values using the known sine values, resulting in:
  \( (1, 0) \), \( (2, \frac{\pi}{2}) \), \( (1, \pi) \), \( (0, \frac{3\pi}{2}) \), \( (1, 2\pi) \).
• Plot these points on a polar grid.
• Connect the plotted points smoothly, following the symmetry and shape progression of the curve.

This systematic approach to plotting ensures the cardioid shape appears as expected, emphasizing the seamless transition of polar coordinates from angle to radius. With practice, spotting these transitions can make polar graphing intuitive and engaging.