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The graph of the polar curve is shown above for . What is the area of the shaded region?
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9. Polar Equations
Polar Coordinate System
Multiple Choice
Given the point with polar coordinates , which of the following polar coordinate pairs represents the same point?
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검증된 단계별 안내1
Recall that polar coordinates are given as \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis.
Understand that the same point in polar coordinates can have multiple representations by adjusting \(r\) and \(\theta\) using these properties: \((r, \theta)\) is the same as \((r, \theta + 2k\pi)\) for any integer \(k\), and also \((r, \theta)\) is the same as \((-r, \theta + (2k+1)\pi)\) for any integer \(k\).
Given the point \((3, \frac{\pi}{4})\), to find an equivalent point with a positive radius, add multiples of \(2\pi\) to the angle \(\theta\). For example, adding \(2\pi\) (which is \(\frac{8\pi}{4}\)) to \(\frac{\pi}{4}\) gives \(\frac{9\pi}{4}\), so \((3, \frac{9\pi}{4})\) represents the same point.
To find an equivalent point with a negative radius, change \(r\) to \(-3\) and add \(\pi\) (or an odd multiple of \(\pi\)) to the angle. For example, \((-3, \frac{\pi}{4} + \pi) = (-3, \frac{5\pi}{4})\) represents the same point.
Check each given option by applying these transformations to see which coordinate pair matches the original point \((3, \frac{\pi}{4})\).
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