The graphs of the polar equations r₁= cos(20) and r₂ = sin(9) + 1 are shown below. Find the 4 angles between 0 and 2 for which r₁=0, and the angle for which r₂ = 0, please explain every steps clearly. Then find the other six points of intersection (r and 9) of the curves. You should be able to find four values of r and exactly, and find the other two to four decimal places. Hint: r₁ negative values, while r₂ = sin(9) + 1 takes on only non- = cos (20) takes on both positive and negative values.
The graphs of the polar equations r₁= cos(20) and r₂ = sin(9) + 1 are shown below. Find the 4 angles between 0 and 2 for which r₁=0, and the angle for which r₂ = 0, please explain every steps clearly. Then find the other six points of intersection (r and 9) of the curves. You should be able to find four values of r and exactly, and find the other two to four decimal places. Hint: r₁ negative values, while r₂ = sin(9) + 1 takes on only non- = cos (20) takes on both positive and negative values.
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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Transcribed Image Text:The graphs of the polar equations \( r_1 = \cos(2\vartheta) \) and \( r_2 = \sin(\vartheta) + 1 \) are shown below.
Find the 4 angles between 0 and \( 2\pi \) for which \( r_1 = 0 \), and the angle for which \( r_2 = 0 \). Please explain every step clearly. Then find the other six points of intersection (\( r \) and \(\vartheta\)) of the curves. You should be able to find four values of \( r \) and \(\vartheta\) exactly, and find the other two to four decimal places. Hint: \( r_1 = \cos(2\vartheta) \) takes on both positive and negative values, while \( r_2 = \sin(\vartheta) + 1 \) takes on only non-negative values.
**Graph Description:**
The graph displays two polar curves:
1. The blue curve, representing \( r_2 = \sin(\vartheta) + 1 \), forms a larger circular shape.
2. The red curve, representing \( r_1 = \cos(2\vartheta) \), forms a four-petaled rose with lobes symmetric about the origin.
Both curves intersect at several points, which are the focus of the exercise. The polar plot is centered at the origin, with four quadrants marked and grid lines shown for reference.
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