Problem 35 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identify and sketch the graph of the polar equation. Identify any symmetry and zeros of \(r .\) Use a graphing utility to verify your results. $$r=1+2 \sin \theta$$

Step-by-Step Solution

Answer

The polar graph of the equation \(r=1+2 \sin \theta\) does not exhibit any symmetry. The zeros of \(r\) are at \(\theta = 7\pi/6, 11\pi/6\). The plot of the equation creates a cardioid like curve.

Step 1: Identifying Symmetry

First of all, verify the symmetry of the given equation. An equation is symmetrical about the x-axis if \((-r, -\theta)\) is identical to \((r,\theta)\), y-axis if \((-r, \theta)\) equals \((r, -\theta)\), or origin if \((-r, -\theta)\) is equal to \((r, \theta)\). For the given equation \(r=1+2 \sin \theta\), there is no such \(\theta\) that will give us negative \(r\). Therefore, this equation has no symmetry.

Step 2: Finding Zeros of r

Zeros of \(r\) are the points where \(r=0\). This happens when \(1+2 \sin \theta = 0\). Solving it by subtracting 1 from both sides we get \(2 \sin \theta = -1\) or \(\sin \theta = -1/2\). This implies \(\theta = 7\pi/6, 11\pi/6\). Therefore, the zeros of \(r\) are \(\theta = 7\pi/6, 11\pi/6\).

Step 3: Sketching the Graph

You can plot the graph by creating a table of values for \(\theta\) and its corresponding \(r\) and then plotting these points on the polar graph. Specially make sure to check the value of \(r\) for \(\theta = 0, \pi/2, \pi, 3\pi/2\) and \(2\pi\). For \(\theta = 0\) and \(2\pi\), \(r = 1+2\sin(0) = 1\). For \(\theta = \pi/2\), \(r = 1+2\sin(\pi/2) = 3\). For \(\theta = \pi\) and \(3\pi/2\), \(r = 1+2\sin(\pi) = 1\) and \(r = 1+2\sin(3\pi/2) = -1\), but in polar coordinates \(r\) can't be negative, so when \(r\) is negative, transform the coordinates to \((-r, \theta + \pi)\) to get valid polar coordinates. Therefore, when \(\theta = 3\pi/2\), use \(r = 1\) and \(\theta = \pi/2\). Sketch the graph using these points and by following the shape of a sinusoid curve. Lastly, use a graphing utility to verify your results.

Key Concepts

Graphing Polar Equations

When graphing polar equations, it's important to remember that the point of reference is the pole (origin) and the angle is measured in radians from the positive x-axis. The equation given, \(r = 1 + 2 \sin \theta\), is a type of polar curve called a limacon. To graph it, you should determine \(r\) for several values of \(\theta\), such as \(0, \pi/2, \pi, 3\pi/2, \text{and} \; 2\pi\).

For example, at \(\theta = 0\), the sine term disappears making \(r = 1\). At \(\theta = \pi/2\), since \(\sin\;\pi/2 = 1\), \(r\) becomes 3. As you go full circle around to \(\theta = 2\pi\), the graph returns nearby the start point. This periodic nature of sine aligns the shape in a loop. The values can be plotted and connected smoothly following the general wave pattern set by sine. Reading these values on a polar plot truly helps in visualizing the equation.

Symmetry in Polar Graphs

Symmetry in polar graphs can help in predicting and sketching the graph more accurately. There are three types of symmetry to check: symmetry about the x-axis, y-axis, and origin. For our equation \(r = 1 + 2 \sin \theta\), checking these symmetries can indicate redundant areas of the graph.

To identify if there is x-axis symmetry, substitute \((-r, -\theta)\) for \((r, \theta)\). For y-axis symmetry, substitute \((-r, \theta)\) with \((r, -\theta)\). For origin symmetry, check \((-r, -\theta)\) with \((r, \theta)\) again, but with adjustments for negative \(r\). In this case, none of those tests uphold any symmetry for the given equation, so there's no evident x-axis, y-axis, or origin symmetry. Having no symmetry does not affect graphing negatively but requires plotting the full set for accuracy.

Zeros of Polar Functions

Finding zeros in polar functions is a critical step. Zeros of \(r\) are points where \(r = 0\), meaning the point lies at the pole. Solving \(r = 1 + 2 \sin \theta = 0\) gives \(\sin \theta = -1/2\).

This equation signals two zeros of \(r\), happening at angles \(\theta = 7\pi/6\) and \(11\pi/6\). On a unit circle or polar graph, these angles show where the curve intersects the pole, denoting them as critical points for graphing.

These zeros assist in sketching the polar graph correctly, indicating points to draw the curve through the pole. Understanding where zeros occur enriches the plotting process, ensuring all major features of the curve are visible and accounted for in the sketch.