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Magazine Chapter 9: Problem 13
Graph the polar equations. $$r=2-2 \sin \theta$$
Short Answer
Expert verified
The graph of \( r = 2 - 2 \sin \theta \) is a cardioid.
Step by step solution
01
Understanding Polar Coordinates
Polar coordinates represent points in a plane by an angle and a radius. Here, \( r \) is the radius, and \( \theta \) is the angle (in radians) from the positive x-axis. The equation \( r = 2 - 2 \sin \theta \) defines the radius as a function of the angle \( \theta \).
02
Identifying the Type of Graph
The given equation \( r = 2 - 2 \sin \theta \) often represents a limacon. Based on the form \( r = a - b \sin \theta \), where \( a = 2 \) and \( b = 2 \), the graph will be a cardioid because \( a = b \).
03
Finding Key Points
To graph the equation, calculate \( r \) for key values of \( \theta \). For example, when \( \theta = 0 \), \( \sin 0 = 0 \), so \( r = 2 \). When \( \theta = \frac{\pi}{2} \), \( \sin \frac{\pi}{2} = 1 \), hence \( r = 0 \).
04
Plotting the Points
Plot the points corresponding to the calculated \( r \) for given \( \theta \) values on polar axis. Extend calculations for points like \( \theta = \pi, \theta = \frac{3\pi}{2} \) and so on.
05
Drawing the Graph
Join the plotted points smoothly, considering the symmetry. The graph will form a cardioid which looks somewhat like a heart, starting from the pole, looping outwards, and returning to the pole.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polar Equations
Polar equations are a way to define curves on a two-dimensional plane using a radius and an angle. This approach is different from the Cartesian coordinate system, which uses x and y coordinates. In polar coordinates, each point is defined by:
- Radius (\( r \)) - the distance from the origin (pole).
- Angle (\( \theta \)) - the angular direction from the positive x-axis.
Graphing Polar Curves
Graphing polar curves involves plotting points on a polar coordinate grid by calculating values for \( r \) for various angles \( \theta \). Here are the steps:
- Start by choosing key angles such as \( 0, \frac{\pi}{2}, \pi, \text{and} \frac{3\pi}{2} \).
- Calculate \( r \) for each angle using your polar equation.
- Plot each calculated point on a polar grid where \( \theta \) is the angle line, and \( r \) is the distance from the origin.
Cardioid Graph
A cardioid is a specific type of polar curve that resembles the shape of a heart. Polar equations of the form \( r = a - b \sin \theta \) or \( r = a - b \cos \theta \) often generate this shape. When \( a = b \), the equation describes a perfect cardioid.
In our example, \( r = 2 - 2 \sin \theta \) fulfills this condition because \( a = b = 2 \). This means the curve will:
In our example, \( r = 2 - 2 \sin \theta \) fulfills this condition because \( a = b = 2 \). This means the curve will:
- Begin at a point on the polar axis, loop outward, creating a cusp at the pole,
- and return back to the starting point.
Understanding Polar Coordinates
Understanding polar coordinates is crucial for navigating problems involving circular and spiraled shapes. Unlike Cartesian coordinates, polar coordinates provide a different perspective:
- Origin-Centric: Points revolve around an origin, making it suitable for circular patterns.
- Angle and Radius: Defining points by angle (\( \theta \)) and distance (\( r \)) offers unique insights into radial symmetry and rotations.
