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Magazine Chapter 9: Problem 19
15鈥�36 Sketch the graph of the polar equation. $$r=6 \sin \theta$$
Short Answer
Expert verified
The graph is a circle with radius 3, centered at (0,3) on the polar plane.
Step by step solution
01
Understanding the Polar Equation
The given polar equation is \( r = 6 \sin \theta \). In polar coordinates, \( r \) represents the radial distance from the origin to a point, and \( \theta \) is the angle from the positive x-axis. This particular equation is a sinusoidal function in terms of \( \theta \).
02
Identify the Type of Graph
For polar equations of the form \( r = a \sin \theta \), the graph represents a circle. Specifically, it will be a circle whose center is along the polar axis (y-axis in Cartesian coordinates) and will be above the horizontal axis since \( \sin \theta \) is non-negative for \( \theta \) from \( 0 \) to \( \frac{\pi}{2} \).
03
Determine Key Points
We determine important points by considering significant \( \theta \) values: \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \).- \( \theta = 0: r = 6 \sin 0 = 0 \).- \( \theta = \frac{\pi}{2}: r = 6 \sin \frac{\pi}{2} = 6 \).- \( \theta = \pi: r = 6 \sin \pi = 0 \).- \( \theta = \frac{3\pi}{2}: r = 6 \sin \frac{3\pi}{2} = -6 \).
04
Sketching the Polar Graph Based on Points
Start plotting the determined points in the polar plane. At \( \theta = 0 \) and \( \theta = \pi \), the point is located at the origin (\( r = 0 \)). At \( \theta = \frac{\pi}{2} \), the graph peaks at \( r = 6 \) (6 units above the origin). At \( \theta = \frac{3\pi}{2} \), reflect below to -6, making a full circle.
05
Drawing the Circle
Connect these points smoothly to form a circle with a diameter along the y-axis, centered at (0,3) with a radius of 3. The circle is entirely above the x-axis, and its lowest point is the origin, with the upper point being (0,6).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sinusoidal Function
A sinusoidal function is a mathematical function that describes a smooth, periodic oscillation. In this exercise, the polar equation \( r = 6 \sin \theta \) is a sinusoidal function with respect to \( \theta \). It scales the basic sine function by a factor of 6, meaning the amplitude or height of the function is 6 units from the midline. This function oscillates between 6 and -6, generating values for \( r \) that guide the shape of the graph.
- Periodic Nature: The sine function repeats every \( 2\pi \) radians, creating a cycle of angles that define points on the circle graph.
- Amplitude Change: The factor of 6 amplifies the sine wave, extending the circle's radius.
Radial Distance
In polar coordinates, the radial distance \( r \) represents how far a point is from the origin (or pole). For the equation \( r = 6 \sin \theta \), the radial distance is determined by the sine of the angle \( \theta \) multiplied by 6. As \( \theta \) changes, \( r \) oscillates between 6 and -6, guiding how far from the origin a point sits.
- Positive Radial Values: When \( \sin \theta \) is positive, \( r \) is positive, indicating that the point is in the direction defined by \( \theta \).
- Negative Radial Values: When \( \sin \theta \) results in a negative \( r \), it reflects along the polar axis (180-degree shift).
Circle Graph
The graph of the equation \( r = 6 \sin \theta \) forms a circle in polar coordinates. For this particular setup, changes in \( \theta \) trace out a circular shape due to the sinusoidal nature of \( r \). The circle's center isn't at the origin but shifted along the polar axis.
- Graph Position: The circle lies centered at the origin's y-axis in Cartesian terms, but this results from the characteristics of \( \sin \theta \).
- Circle Characteristics: The circle's diameter is 6 units, with points reaching their maximum at \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \).
Angle Measurement
In polar coordinates, angle measurement is crucial for determining the position of points around the origin. The angle \( \theta \) in \( r = 6 \sin \theta \) is measured from the positive x-axis, dictating where the point lands on the polar plane. Understanding these angles assists in plotting points accurately.
- Quadrantal Angles: Strategic angles like \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \) are used to calculate key radial distances, forming the structure of the polar graph.
- Range of Angles: The angles dictate the periodic rise and fall of \( r \), commanding the design of the sinusoidal wave and subsequently, the circle graph.
