Question

For each of the points given in polar coordinates, find two additional pairs of polar coordinates $(r, \theta),$ one with $r>0$ and one with $r<0$. $$\left(4, \frac{3 \pi}{2}\right)$$

   For each of the points given in polar coordinates, find two additional pairs of polar coordinates $(r, \theta),$ one with $r>0$ and one with $r<0$.
$$\left(4, \frac{3 \pi}{2}\right)$$

Precalculus : Building Concepts and Connections

Precalculus : Building Concepts and Connections

Revathi Narasimhan 1st Edition

Chapter 7, Problem 35

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Solved by Expert Linh Vu

04/13/2020

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We know that we can add or subtract $2\pi$ to the angle $\theta$ to get an equivalent point in polar coordinates. Show more…

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For each of the points given in polar coordinates, find two additional pairs of polar coordinates $(r, \theta),$ one with $r>0$ and one with $r<0$. $$\left(4, \frac{3 \pi}{2}\right)$$

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Key Concepts

Polar Coordinates

Polar coordinates are a method of representing points in the plane using a distance from a fixed reference point (the origin) and an angle measured from a fixed reference direction (usually the positive x?axis). This system allows for a concise description of points using a pair (r, ?), where r is the radial distance and ? is the angular coordinate.

Equivalent Representations

In the polar coordinate system, the same point can be represented by multiple coordinate pairs due to the periodic nature of the angle and the possibility of using a negative radial distance. For instance, adding multiples of 2? to the angle produces an equivalent point, and a coordinate pair with r negative can be converted into an equivalent pair with r positive by adding ? to the angle.

Handling Negative Radii

A negative radius in polar coordinates essentially points in the direction opposite to the specified angle. To convert a pair with a negative r into one with a positive r, you add ? (or 180°) to the angle. This adjustment realigns the point so that it corresponds to the correct position in the plane, ensuring that the representation remains accurate.

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