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9. Sketch the polar curves r = 5sine and r = 5 sin(30) on the same grid below.

9. Sketch the polar curves r = 5sine and r = 5 sin(30) on the same grid below.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Sketch the polar curves ? = 5???? ??? ? = 5 sin(3?) ?? ?ℎ? ???? ???? ?????.

**Exercise: Sketching Polar Curves** **Task:** Sketch the polar curves \( r = 5\sin\theta \) and \( r = 5\sin(3\theta) \) on the same grid below. **Graph Explanation:** The provided grid is a polar coordinate grid consisting of concentric circles centered at the origin, intersected by straight lines radiating from the center. This type of graph is used to plot points in polar coordinates where each point on the plane is determined by an angle and a distance from the origin. - The radial lines are equally spaced at angles, usually representing increments of typical polar angles (in degrees or radians). - The concentric circles typically represent units of radial distance from the center, helping to gauge the radius of plotted curves. To sketch the given polar curves, you need to plot the points for these equations: 1. **For \( r = 5\sin\theta \):** - This is a sinusoidal curve that has maximum radius \( r = 5 \) and will appear as a circle since \( \sin\theta \) ranges from -1 to 1. 2. **For \( r = 5\sin(3\theta) \):** - This curve creates a rose with multiple petals, as the factor \( 3\theta \) results in a tripling effect of the angles, typically resulting in three petals for positive sine functions. **Procedure:** - Determine key points where \(\theta\) takes on values (such as \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{2}, \pi\), etc.) and calculate \( r \). - Plot these points on the grid and join them smoothly to visualize the curves. These exercises will help in understanding the behavior of polar equations and foster visual spatial understanding of radial symmetry and periodicity in polar graphs.
Transcribed Image Text:**Exercise: Sketching Polar Curves** **Task:** Sketch the polar curves \( r = 5\sin\theta \) and \( r = 5\sin(3\theta) \) on the same grid below. **Graph Explanation:** The provided grid is a polar coordinate grid consisting of concentric circles centered at the origin, intersected by straight lines radiating from the center. This type of graph is used to plot points in polar coordinates where each point on the plane is determined by an angle and a distance from the origin. - The radial lines are equally spaced at angles, usually representing increments of typical polar angles (in degrees or radians). - The concentric circles typically represent units of radial distance from the center, helping to gauge the radius of plotted curves. To sketch the given polar curves, you need to plot the points for these equations: 1. **For \( r = 5\sin\theta \):** - This is a sinusoidal curve that has maximum radius \( r = 5 \) and will appear as a circle since \( \sin\theta \) ranges from -1 to 1. 2. **For \( r = 5\sin(3\theta) \):** - This curve creates a rose with multiple petals, as the factor \( 3\theta \) results in a tripling effect of the angles, typically resulting in three petals for positive sine functions. **Procedure:** - Determine key points where \(\theta\) takes on values (such as \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{2}, \pi\), etc.) and calculate \( r \). - Plot these points on the grid and join them smoothly to visualize the curves. These exercises will help in understanding the behavior of polar equations and foster visual spatial understanding of radial symmetry and periodicity in polar graphs.
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