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Find all points (if any) of horizontal and vertical tangency to the portion of the curve shown. (Enter your answers as a comma-separated list.) x = cos 0 + 0 sin y sin 8-0 cos Ⓒ -2n S 0 s 2n 0-7,2 horizontal tangents vertical tangents 8 M ++ -8-6 7 - л Зл Зл

Find all points (if any) of horizontal and vertical tangency to the portion of the curve shown. (Enter your answers as a comma-separated list.) x = cos 0 + 0 sin y sin 8-0 cos Ⓒ -2n S 0 s 2n 0-7,2 horizontal tangents vertical tangents 8 M ++ -8-6 7 - л Зл Зл

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem Statement:** Find all points (if any) of horizontal and vertical tangency to the portion of the curve shown. (Enter your answers as a comma-separated list.) **Parametric Equations:** - \( x = \cos \theta + \theta \sin \theta \) - \( y = \sin \theta - \theta \cos \theta \) - \(-2\pi \leq \theta \leq 2\pi \) **Horizontal Tangents:** \[ \theta = \pi, 2\pi \] **Vertical Tangents:** \[ \theta = \frac{\pi}{2}, -\frac{\pi}{2}, \frac{3\pi}{2}, -\frac{3\pi}{2} \] **Explanation of the Graph:** The graph features a polar curve in the XY-plane. The graph is centered on the origin and extends outward in a circular pattern, showing symmetry around the x-axis. The plotted curve undulates around the center, suggesting that at angles listed for horizontal tangents (\(\pi, 2\pi\)), the slope of the curve is 0, meaning the curve is flat and tangent to the x-axis. At the angles given for vertical tangents (\(\frac{\pi}{2}, -\frac{\pi}{2}, \frac{3\pi}{2}, -\frac{3\pi}{2}\)), the curve is tangent to lines perpendicular to the x-axis, indicating infinite slope. The graph helps visualize these specific points of tangency.
Transcribed Image Text:**Problem Statement:** Find all points (if any) of horizontal and vertical tangency to the portion of the curve shown. (Enter your answers as a comma-separated list.) **Parametric Equations:** - \( x = \cos \theta + \theta \sin \theta \) - \( y = \sin \theta - \theta \cos \theta \) - \(-2\pi \leq \theta \leq 2\pi \) **Horizontal Tangents:** \[ \theta = \pi, 2\pi \] **Vertical Tangents:** \[ \theta = \frac{\pi}{2}, -\frac{\pi}{2}, \frac{3\pi}{2}, -\frac{3\pi}{2} \] **Explanation of the Graph:** The graph features a polar curve in the XY-plane. The graph is centered on the origin and extends outward in a circular pattern, showing symmetry around the x-axis. The plotted curve undulates around the center, suggesting that at angles listed for horizontal tangents (\(\pi, 2\pi\)), the slope of the curve is 0, meaning the curve is flat and tangent to the x-axis. At the angles given for vertical tangents (\(\frac{\pi}{2}, -\frac{\pi}{2}, \frac{3\pi}{2}, -\frac{3\pi}{2}\)), the curve is tangent to lines perpendicular to the x-axis, indicating infinite slope. The graph helps visualize these specific points of tangency.
Expert Solution
Step 1: Given

The equation of the given curve is 

x equals cos theta plus theta sin theta y equals sin theta minus theta cos theta minus 2 pi less or equal than theta less or equal than 2 pi

To find all the points or theta values where the tangents are either vertical or horizontal.


The slope of the tangent of a curve is represented by m equals fraction numerator d y over denominator d x end fraction, 

if the tangent is horizontal then its slope is zero,

if the tangent is vertical then the slope is undefined or fraction numerator d x over denominator d theta end fraction equals 0 .


It is known that tan open parentheses n pi close parentheses equals 0 for n element of straight integer numbers,

and tan open parentheses fraction numerator n pi over denominator 2 end fraction close parentheses for n element of straight integer numbers undefined.


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