Use a polar coordinate system like the one shown for...

Use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. Then find another representation (r, $\theta$) of this point in which. $$\text { a. } r>0, \quad 2 \pi<\theta<4 \pi$$ $$\text { b. } r<0, \quad 0<\theta<2 \pi$$ $$\text { c. } r>0, \quad-2 \pi<\theta<0$$ $$ \left(4, \frac{\pi}{2}\right) $$

Use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. Then find another representation (r, $\theta$) of this point
in which.
$$\text { a. } r>0, \quad 2 \pi<\theta<4 \pi$$
$$\text { b. } r<0, \quad 0<\theta<2 \pi$$
$$\text { c. } r>0, \quad-2 \pi<\theta<0$$
$$
\left(4, \frac{\pi}{2}\right)
$$

Key Concepts

Negative Radius and Angle Adjustment

Using a negative radius in polar coordinates is equivalent to using the positive radius with the angle shifted by ? radians. This method of adjustment allows for flexibility in how a point is represented, enabling one to convert a representation with a positive radius to one with a negative radius (or vice versa) by appropriately modifying the angle, to meet the constraints of a particular problem.

Angle Periodicity

The periodicity of angles is a fundamental concept in polar coordinates, meaning that angles differing by full rotations (2? radians) correspond to the same terminal direction. This idea is leveraged to adjust the angle in a polar coordinate pair to satisfy given constraints, such as a specific interval, while still representing the same point.

Equivalent Representations in Polar Coordinates

A single point in the plane can have multiple valid polar coordinate representations because of the periodicity of angles. By adding or subtracting full rotations (typically multiples of 2? radians), or by changing the sign of the radius and adjusting the angle accordingly, one can convert among these equivalent representations, ensuring the point remains the same in the Cartesian plane.

Polar Coordinates

Polar coordinates provide a way to represent points in the plane using a distance from a fixed origin and an angle measured from a fixed direction. This system is especially useful for situations where radial symmetry is present or the relationships between points are more naturally expressed in terms of distance and direction rather than horizontal and vertical displacements.

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