# How do polar coordinates work?

Please see the video for Polar Coordinate basics.

Polar Coordinate Basics

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Polar coordinates represent points in a plane using a distance from a reference point (the pole or origin) and an angle from a reference direction (usually the positive x-axis). The distance is denoted by ( r ) and the angle by ( \theta ). To convert from Cartesian coordinates (x, y) to polar coordinates (r, ( \theta )), we use the formulas:

[ r = \sqrt{x^2 + y^2} ] [ \theta = \arctan\left(\frac{y}{x}\right) ]

To convert from polar coordinates to Cartesian coordinates:

[ x = r \cos(\theta) ] [ y = r \sin(\theta) ]

Polar coordinates are particularly useful in describing circular or radial patterns, such as in physics, engineering, and navigation. They provide a natural way to describe points relative to a central point and an angle of rotation.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- What is the polar form of #( -4,5 )#?
- How do you sketch the graph of the polar equation and find the tangents at the pole of #r=2(1-sintheta)#?
- How do you find the area of the common interior of #r=4sintheta, r=2#?
- What is the arclength of the polar curve #f(theta) = -3sin(3theta)-2cot4theta # over #theta in [0,pi/8] #?
- What is the Cartesian form of #( 7 , ( 7pi)/3 ) #?

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