Problem 40 - Algebra and Trigonometry

Question

Identify and graph each polar equation. $$ r=1+\sin \theta $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Polar Equation

The given polar equation is \( r = 1 + \sin\theta \). This is a polar equation where the radius \( r \) is expressed in terms of the angle \( \theta \).

Step 2: Identify the Type of Polar Graph

This equation \( r = 1 + \sin\theta \) represents a limaçon. Limaçons can have different shapes including dimpled, cardioid, or with an inner loop, depending on the coefficients.

Step 3: Determine Key Points

To plot the graph, calculate the value of \( r \) for key angles (like \( 0 \), \( \frac{\pi}{2} \), \( \pi \), and \( \frac{3\pi}{2} \)). For instance, at \( \theta = 0 \), \( r = 1 + \sin(0) = 1 \). Repeat this process for other angles to find the corresponding \( r \) values.

Step 4: Plot the Points

Using the key points obtained, plot those (r, \( \theta \)) points on polar coordinate paper. For example, at \( \theta = 0 \) and \( r = 1 \), plot a point 1 unit away from the origin along the polar axis. Do the same for other calculated points.

Step 5: Draw the Polar Graph

Connect the points plotted smoothly to illustrate the shape of the limaçon. Note that the graph will form a dimpled shape because the amplitude is not large enough compared to the constant term.

Step 6: Conclusion

The polar equation \( r = 1 + \sin \theta \) represents a dimpled limaçon which is plotted based on the calculated key points and their connection.

Key Concepts

Limaçon

A limaçon is a special type of polar graph that presents interesting and varied shapes. The term 'limaçon' comes from the Latin word for 'snail', reflecting its curved structure. A limaçon is described by the polar equation of the form \( r = a + b \times \text{sin}(\theta) \) or \( r = a + b \times \text{cos}(\theta) \).

The shapes of limaçons can vary greatly:

• **Cardioid**: When \( a = b \), the limaçon looks like a heart and is called a cardioid.
• **Dimpled Limaçon**: When \( 0 < b < a \), the limaçon has a dimple.
• **Convex Limaçon**: When \( b < a \) and there is no dimple.
• **Inner Loop**: When \( b > a \), the limaçon has an inner loop.

In the given problem, the equation \( r = 1 + \text{sin}(\theta) \) forms a dimpled limaçon since the coefficient of \(\text{sin}(\theta)\) is less than the constant term. This creates a unique and illustrative shape around the origin.

Polar Coordinates

Polar coordinates are a system of defining points on a plane using two values: the radius \( r \) and the angle \( \theta \).

Instead of using Cartesian coordinates \( (x, y) \), polar coordinates allow us to describe a point's location by how far it is from the origin (radius) and the direction (angle) from a reference direction, usually the positive x-axis.

Here's a quick breakdown:

• **Radius (r)**: The distance from the origin to the point.
• **Angle (\( \theta \))**: The angle measured from the positive x-axis to the radius line, usually in radians or degrees.

Mapping polar coordinates on polar graph paper, you can easily plot points by measuring the angle \( \theta \) and moving along the reference line by the radial distance \( r \).

In the problem given, key angles like \( 0 \), \( \frac{\pi}{2} \), \( \pi \), and \( \frac{3\pi}{2} \) are used to compute the corresponding \( r \) values and plot the points.

Graphing Polar Equations

Graphing polar equations involves plotting points and connecting them based on relationships defined by the polar equations.

Let's walk through the process:

• **Identify Key Angles**: Begin by selecting crucial angles \( \theta \) which simplify the computation of \( r \). Examples include \( \theta = 0 \), \( \frac{\pi}{2} \), \( \pi \), and \( \frac{3\pi}{2} \).
• **Calculate Corresponding \( r \) values**: Substitute each selected angle into the equation to find \( r \). For instance, with \( \theta = 0 \), the given equation \( r = 1 + \text{sin}(\theta) \) yields \( r = 1 \).
• **Plot Points**: On polar coordinates, use the angle \( \theta \) and the computed radius \( r \) to mark points.
  - For \( \theta = 0 \) and \( r = 1 \), place a point 1 unit from the origin.
  - Repeat this for other angles.
• **Connect Points**: Join the points smoothly to form the graph. The shape will visually represent the type of curve, like a limaçon.

This method helps in visualizing complex relationships in polar equations. The practice on graph paper aids in understanding the equation's behavior and the resulting shape.