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Polar Coordinates

Polar coordinates are a way to describe where a point is on a plane. Instead of using x and y, you use the angle theta and radius r, to describe the angle and distance of the point from the origin.

Converting Polar Coordinates to Cartesian

         

The point \( \left(12, \frac{3\pi}{4} \right)\) in polar coordinates can be expressed as \((a, b)\) in Cartesian coordinates, where \(a\) and \(b\) are real numbers. What is the value of \(-ab?\)

Suppose that \((a, \frac{\pi}{6})\) in polar coordinates represents the same point as \((b, 7)\) in Cartesian coordinates. What is the value of \(a^2+b^2\)?

\(P=(24, \frac{\pi}{3})\) in polar coordinates can be expressed as \(P=(a, b)\) in Cartesian coordinates. What is the value of \(a+b^2\)?

If point \(P\) is given in polar coordinates as \(P=\left(-10, \frac{3\pi}{4}\right),\) what are the Cartesian coordinates of \(P?\)

In polar coordinates, the graph of \(r = 46 \sin(\theta)\) is a circle. What is the radius of this circle?

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