Question

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

Short answer

The graph is a cardioid symmetric about the vertical axis.

Step-by-step solution

Identify Type of Polar Graph

The given polar equation is \( r = 1 + \sin \theta \). This is a limacon with an inner loop since the equation is in the form \( r = a + b \sin \theta \) and \( b = a = 1 \). Limacons can have different shapes depending on the values of \( a \) and \( b \). Here, since \( a = b \), we suspect a cardioid.

Determine Symmetry

A quick way to check for symmetry in polar graphs is to look for specific trigonometric functions. Since this equation includes \( \sin \theta \), it tends to be symmetric about the line \( \theta = \frac{\pi}{2} \) (vertical axis symmetry). We confirm this by substitution. If \( \theta \) changes to \(\pi - \theta \), \( \sin \theta = \sin (\pi - \theta) \), thus proving symmetry about the vertical axis.

Calculate Key Points

To sketch the graph, we calculate a few key values. For \( \theta = 0 \), \( r = 1 + \sin 0 = 1 \). For \( \theta = \frac{\pi}{2} \), \( r = 1 + \sin \frac{\pi}{2} = 2 \). For \( \theta = \pi \), \( r = 1 + \sin \pi = 1 \). For \( \theta = \frac{3\pi}{2} \), \( r = 1 + \sin \frac{3\pi}{2} = 0 \). The points \((1, 0)\), \((2, \frac{\pi}{2})\), \((1, \pi)\), and \((0, \frac{3\pi}{2})\) are critical to sketching the limacon.

Sketch the Graph

Using the calculated points, begin by plotting these on polar graph paper. Start at \((1, 0)\), move to \((2, \frac{\pi}{2})\), back down to \((1, \pi)\), and finally to the pole (origin) at \((0, \frac{3\pi}{2})\). Connect these smoothly, taking care to loop around and create the heart-shaped cardioid, showing the loop protusion between \( (0, \frac{3\pi}{2}) \) and back up to the initial points.

Key concepts

Understanding Limacons

Polar coordinates are a system where each point on a plane is determined by an angle and a distance. A limacon is a unique type of graph that you can encounter with polar coordinates. It is typically described by the equation \( r = a + b \sin(\theta) \) or \( r = a + b \cos(\theta) \). Depending on the relationship between \( a \) and \( b \), limacons can take various shapes.

• If \( a > b \), the limacon has no inner loop and resembles a distorted circle.
• If \( a < b \), it features an inner loop, portraying a more complex shape.
• When \( a = b \), the limacon transforms into a cardioid, a peach-shaped curve.

Understanding these forms is essential as they guide how you interpret and sketch the graph visually. The graph of \( r = 1 + \sin(\theta) \) falls into the category where \( a = b \), so it becomes a special case of limacon known as the cardioid.

Identifying the Cardioid

A cardioid is a type of limacon and it gets its name due to its heart-like shape. The standard form of a cardioid in polar coordinates is \( r = a + a \sin(\theta) \) or \( r = a + a \cos(\theta) \).
In the exercise, the equation \( r = 1 + \sin(\theta) \) aligns perfectly with the cardioid form where \( a = 1 \). The cardioid's unique shape is due to the circular arc folding inward at one point, creating the characteristic cusp. This shape is crucial in applications like antenna design, where signal propagation may utilize such symmetrical patterns.
For another visual perspective, imagine this: as \( \theta \) varies from \( 0 \) to \( 2\pi \), the distance \( r \) varies smoothly creating the curve that hugs the axis and intersects it again, completing the cardioid loop.

Graph Symmetry in Polar Coordinates

Symmetry in graphs allows for simplified sketching and analysis. Polar graphs exhibit specific kinds of symmetry based on their structure.

• **Symmetry about the polar axis (horizontal):** This is often seen in equations with \( \cos(\theta) \).
• **Symmetry about the line \( \theta = \frac{\pi}{2} \) (vertical):** Most common in \( \sin(\theta) \) equations like \( r = 1 + \sin(\theta) \). The values of \( r \) will reflect over this line, creating a mirror image.
• **Symmetry about the origin:** If replacing \( (r, \theta) \) with \( (-r, \theta + \pi) \) results in the same equation.

For the problem at hand, vertical symmetry is proven through the trigonometric property that \( \sin(\theta) = \sin(\pi - \theta) \). This symmetry simplifies the plotting process since you can mirror the points found in one-half of the graph to the other half, ensuring a perfect cardioid shape on polar plots.