Question

Graph the polar function on the given interval. \(r=1+2 \sin \theta, \quad[0,2 \pi]\)

Short answer

The graph of the equation is a limaçon with an inner loop.

Step-by-step solution

Understand the Polar Function

The given polar function is represented as \(r = 1 + 2\sin\theta\). In polar coordinates, \(r\) describes the radial distance from the origin, and \(\theta\) is the angle from the positive x-axis. Here, the function will be graphed over the interval \( [0, 2\pi] \), which represents a full rotation.

Determine Key Points

Calculate values of \(r\) for key angles where \( \sin \theta \) has notable values: \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \text{and} \, 2\pi\). For these angles, \(\sin\theta = 0, 1, 0, -1, 0\) respectively. Plug these into the equation to find \(r\):- \(\theta = 0\), \(r = 1 + 2\cdot0 = 1\)- \(\theta = \frac{\pi}{2}\), \(r = 1 + 2\cdot1 = 3\)- \(\theta = \pi\), \(r = 1 + 2\cdot0 = 1\)- \(\theta = \frac{3\pi}{2}\), \(r = 1 + 2\cdot(-1) = -1\)- \(\theta = 2\pi\), \(r = 1 + 2\cdot0 = 1\)

Plot the Key Points

Plot these key points in polar coordinates:- At \(\theta = 0\), the point is at (1,0).- At \(\theta = \frac{\pi}{2}\), the point is at (3,\(\frac{\pi}{2}\)).- At \(\theta = \pi\), the point is at (1,\(\pi\)).- At \(\theta = \frac{3\pi}{2}\), the point is at (-1,\(\frac{3\pi}{2}\)), which is equivalent to 'going backward' to the point (1,\(\frac{\pi}{2}\)).- At \(\theta = 2\pi\), the point is at (1,0), completing the cycle back to the start.

Interpret the Shape

Notice that for \(\theta\) ranging from 0 to \(2\pi\), the function describes a limaçon with an inner loop when \(r\) becomes negative (between \(\pi\) and \(2\pi\)). The graph should display a loop towards the bottom, indicating symmetry about the line \(\theta = \frac{3\pi}{2}\).

Complete the Graph

Using symmetry and the calculated points, sketch the limaçon. Start from \(\theta = 0\), move to \(\theta = \frac{\pi}{2}\), and complete the loop through \(\theta = \pi\) to \(\theta = 2\pi\). The graph will look heart-shaped with a loop below the x-axis.

Key concepts

Polar Functions

Polar functions describe relationships between the radial distance from the origin and the angle from the positive x-axis in a plane. In contrast to the Cartesian system, where we use x and y coordinates, polar coordinates use \( r \) (the radial distance) and \( \theta \) (the angular coordinate).
A polar function can be represented as \( r = f(\theta) \), where \( f \) is a function of \( \theta \). The example \( r = 1 + 2\sin\theta \) specifies how the distance \( r \) changes with the angle \( \theta \).
This representation helps us to easily translate complex shapes and curves that might be cumbersome in Cartesian coordinates.

• Polar functions make use of trigonometric functions like sine and cosine to express the curve's distance and rotation.
• These functions can create various curves like circles, spirals, and limaçons based on the parameters and function form.

Understanding these functions will enhance your grasp of geometry in polar coordinates, helping you visualize and solve problems related to curves and graphing.

Graphing Polar Equations

Graphing polar equations involves plotting points based on the function relationship between \( r \) and \( \theta \).
Begin by calculating key points where \( \theta \) values give notable results, particularly where trigonometric values reach peak or trough, such as \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \) and \( 2\pi \).
For the equation \( r = 1 + 2\sin\theta \), crucial steps include:

• Calculate \( r \) values for each key angle.
• Plot these (\( r, \theta \)) pairs on the polar coordinate plane.
• Connect the points smoothly to capture the curvature of the function.

In our example, this polar function forms a limaçon, an interesting shape often characterized by loops. The inner loop occurs whenever \( r \) becomes negative, as seen when plotting from \( \pi \) to \( 2\pi \).
This function depicts symmetry and periodicity, essential characteristics for interpreting polar graphs effectively.

Trigonometric Functions

Trigonometric functions are foundational in both polar and Cartesian coordinates. These functions, including sine, cosine, and tangent, express relationships between angles and lengths in right-angled triangles.
In polar graphs, trigonometric functions determine how \( r \) changes in relation to \( \theta \). For instance, in \( r = 1 + 2\sin\theta \), the \( \sin\theta \) component dictates how much \( r \) extends or retracts from the baseline value of 1.

• Sine and cosine functions oscillate between -1 and 1, providing rhythmic and predictable variations in \( r \).
• The function's periodic nature aids in predicting its behavior over any interval \( [0, 2\pi] \).
• In practice, alterations in amplitude and phase shift translate directly to changes in the shape and orientation of the graph.

By mastering trigonometric functions, you will not only excel in graphing polar equations but also gain analytical prowess in solving a broad spectrum of mathematical problems.