Polar Coordinates

Plot the following points, given in polar coordinates. Then find all the polar coordinates of each point.

a. (2, π/2)

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Understand that a point in polar coordinates is given as \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle measured from the positive x-axis.

For the point \((2, \frac{\pi}{2})\), plot it by moving 2 units from the origin in the direction of the angle \(\frac{\pi}{2}\) radians (which corresponds to 90 degrees, pointing straight up along the positive y-axis).

To find all polar coordinates for this point, recall that polar coordinates are not unique. You can add or subtract multiples of \(2\pi\) to the angle \(\theta\) without changing the point, so one set of coordinates is \((2, \frac{\pi}{2} + 2k\pi)\) for any integer \(k\).

Also, consider that if you take the negative radius \(-r\), you can add \(\pi\) to the angle to represent the same point. So another set of coordinates is \((-2, \frac{\pi}{2} + (2k+1)\pi)\) for any integer \(k\).

Summarize all possible coordinates as the union of these two sets: \((2, \frac{\pi}{2} + 2k\pi)\) and \((-2, \frac{\pi}{2} + (2k+1)\pi)\), where \(k\) is any integer.

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주요 개념

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Polar Coordinates System

The polar coordinate system represents points in a plane using a radius and an angle. Each point is defined by (r, θ), where r is the distance from the origin and θ is the angle measured from the positive x-axis. This system is useful for describing locations in circular or rotational contexts.

Plotting Points in Polar Coordinates

To plot a point given in polar coordinates, start at the origin, rotate counterclockwise by the angle θ, then move outward along that direction by the distance r. For example, (2, π/2) means move 2 units upward since π/2 radians corresponds to 90 degrees.

Multiple Representations of Polar Coordinates

A single point in the plane can have multiple polar coordinate representations due to angle periodicity and sign of r. Adding 2π to the angle or using a negative radius with an adjusted angle gives equivalent coordinates, so all such forms must be considered when finding all polar coordinates of a point.

Polar CoordinatesPlot the following points, given in polar coordinates. Then find all the polar coordinates of each point.a. (2, π/2)