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Essential Calculus: Early Transcendentals
Found in: Page 528
Essential Calculus: Early Transcendentals

Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

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Short Answer

Find all points of intersection of the given curves.

\({\rm{r = sin\theta ,}}\;\;\;{\rm{r = sin2\theta }}\)

The result of the point of intersection of the curve is pole\({\rm{(}}\sqrt {\rm{3}} {\rm{/2,\pi /3)}}\)and\({\rm{( - }}\sqrt {\rm{3}} {\rm{/2,5\pi /3)}}\).

See the step by step solution

Step by Step Solution

Step 1: Comparing the equation.

Solve\({\rm{\theta }}\) for by making the two equations equal.

\(\begin{aligned}{l}{\rm{sin\theta = sin2\theta }}\\{\rm{sin\theta = 2sin\theta cos\theta }}\end{aligned}\)

Since \({\rm{sin\theta }}\)act as positive and \({\rm{\theta }}\)act as QIV or QI

\(\begin{aligned}{c}{\rm{2sin\theta cos\theta - sin\theta = 0}}\\{\rm{sin\theta (2cos\theta - 1) = 0}}\\{\rm{sin\theta = 0 or cos\theta = 1/2}}\\{\rm{sin\theta = 0}}\\ \Rightarrow {\rm{\theta = 0,\pi and }}\\{\rm{cos\theta = 1/2}}\\ \Rightarrow {\rm{\theta = \pi /3,5\pi /3}}\end{aligned}\)

Step 2: Polar coordinates.

Now plug each value into one of the two equations to get the\({\rm{r}}\)value, and hence the polar coordinates, where the two curves overlap.

\(\begin{aligned}{c}{\rm{\theta = 0}} \Rightarrow {\rm{r = sin0 = 0or}}\\{\rm{r = sin(2 \times 0) = 0}}\\{\rm{\theta = \pi }} \Rightarrow {\rm{r = sin\pi = 0or}}\\{\rm{r = sin(2\pi ) = 0}}\\{\rm{\theta = \pi /3}}\\ \Rightarrow {\rm{r = sin(\pi /3) = }}\sqrt {\rm{3}} {\rm{/2}}\\{\rm{r = sin(2\pi /3) = }}\sqrt {\rm{3}} {\rm{/2}}\\{\rm{\theta = 5\pi /3}}\\ \Rightarrow {\rm{r = sin(5\pi /3)}}\\{\rm{ = - }}\sqrt {\rm{3}} {\rm{/2}}\\{\rm{r = sin(10\pi /3)}}\\{\rm{ = sin(4\pi /3)}}\\{\rm{ = - }}\sqrt {\rm{3}} {\rm{/2}}\\\end{aligned}\)

Therefore, the point of intersection of the curve is pole \({\rm{(}}\sqrt {\rm{3}} {\rm{/2,\pi /3)}}\)and\({\rm{( - }}\sqrt {\rm{3}} {\rm{/2,5\pi /3)}}\).

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