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Polar Equations

Whether you're charting the seas or just doing some plotting in the complex plane, polar equations are the go-to tool for describing points via a distance and angle with respect to the origin.

Polar Coordinates Introduction

         

If the CD is placed on a polar coordinate system with the origin in the center of the hole in the center of the CD, which type of equation does the picture indicate is the worse type of equation for a scratch to approximate?

Microphones are often designed to focus on receiving sounds from certain directions. For example, a cardioid microphone focuses on capturing sound from the front of the microphone, whereas an omnidirectional microphone is designed to capture sound from all directions equally. In the two microphone polar patterns above, a fixed sound pressure level at any point on the graph will produce a constant output level from the microphone at the origin. Which equation would best fit a cardioid microphone's polar pattern?

The equation

\[(x^2 + y^2)^3 = (x^2 - y^2)^2,\]

whose graph is shown, describes a Quadrifolium. What is its equation in polar coordinates?

The picture shows, for \(0 \leq \theta \leq 2\pi,\) the graph of an Archimedian Spiral with equation \[r = \theta,\] and an involute of a circle having equation \[r = \sqrt{1 + \theta^2}.\]

Which is the graph of the Archimedean Spiral?

Chance had some time on his hands and drew this picture using the polar graphing feature of his graphing program. Which answer choice best describes the mouth of this picture?

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