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Chapter 6: Problem 67

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$r=\sin \theta$$

Short Answer

Expert verified
The rectangular equation of \(r = \sin \theta\) is \(y = 1 - x^2\). It represents a downward-opening parabola with vertex at (0,1). The graph cuts the x-axis at (-1,0) and (1,0).

Step by step solution

01

- convert the polar equation to rectangular form

To convert from polar to rectangular, we can use the identities \(x = r \cos \theta\) and \(y = r \sin \theta\). However, in our case \(r = \sin \theta\). So we can write \(y = r \sin \theta\) as \(y = (\sin \theta) \sin \theta\), which reduces to \(y = \sin^2 \theta\). Now \(\sin^2\theta + \cos^2\theta = 1\). So we can write \(\sin^2\theta\) as \(1 - \cos^2\theta\) and the equation will be \(y = 1 - x^2\), where, x = r*cos(theta).
02

- Find the nature of graph

To analyze the graph of \(y = 1 - x^2\), it is observed that it is a downward-opening parabola with vertex at (0,1), that intersects the y-axis at (0,1) and x-axis at (-1,0) and (1,0).
03

- Plotting the graph

Sketch the y-intercept at the point (0,1). Then plot the points (-1,0) and (1,0). Connect these points with a smooth curve to form a downward-opening parabola. This will represent the graph of the given equation \(y = 1 - x^2\).

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Most popular questions from this chapter

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