In Exercises 13–14, graph each polar equation. r = 1 + sin θ

Verified step by step guidance

1. <Step 1: Understand the polar equation. The given equation is $r = 1 + \sin \theta$. This is a polar equation where $r$ is the radius and $\theta$ is the angle in radians.>
2. <Step 2: Identify the type of graph. The equation $r = 1 + \sin \theta$ is a type of limaçon. Specifically, it is a limaçon with an inner loop because the coefficient of $\sin \theta$ is less than the constant term.>
3. <Step 3: Determine key points. Calculate $r$ for key angles $\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$. For example, when $\theta = 0$, $r = 1 + \sin(0) = 1$.>
4. <Step 4: Plot the points. Use the key points calculated in Step 3 to plot the points on the polar coordinate system. For each angle $\theta$, plot the corresponding radius $r$.>
5. <Step 5: Sketch the graph. Connect the plotted points smoothly to form the limaçon shape. Note the inner loop that forms due to the $1 + \sin \theta$ structure.>

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a distance from a reference point (the pole) and an angle from a reference direction (the polar axis). In this system, a point is defined by the coordinates (r, θ), where 'r' is the radial distance and 'θ' is the angle. Understanding polar coordinates is essential for graphing polar equations, as it allows for the visualization of curves that may not be easily represented in Cartesian coordinates.

Polar Equations

Polar equations express relationships between the radial distance 'r' and the angle 'θ'. The equation r = 1 + sin θ describes a curve in polar coordinates, where the value of 'r' changes based on the angle 'θ'. Recognizing how to manipulate and interpret these equations is crucial for accurately graphing the resulting shapes, such as circles or limacons, which can exhibit unique properties depending on the coefficients involved.

Graphing Polar Curves

Graphing polar curves involves plotting points based on the values of 'r' for various angles 'θ'. This process often requires evaluating the equation at key angles (e.g., 0, π/2, π, etc.) to determine the shape and features of the graph. Understanding how to translate the polar equation into a visual representation is vital for interpreting the behavior of the curve, including its symmetry and intersections with the polar axis.