In Exercises 21–26, 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 of this point in which a. r>0, 2π < θ < 4π. b. r<0, 0. < θ < 2π. c. r>0, −2π. < θ < 0. (5, π/6)

Verified step by step guidance

1. Plot the point (5, \pi/6) on the polar coordinate system. This means moving 5 units from the origin at an angle of \pi/6 radians from the positive x-axis.
2. To find another representation with r > 0 and 2\pi < \theta < 4\pi, add 2\pi to the angle: \theta = \pi/6 + 2\pi.
3. To find another representation with r < 0 and 0 < \theta < 2\pi, use the negative radius property: r = -5 and \theta = \pi/6 + \pi.
4. To find another representation with r > 0 and -2\pi < \theta < 0, subtract 2\pi from the angle: \theta = \pi/6 - 2\pi.
5. Verify each new representation by converting back to Cartesian coordinates to ensure they represent the same point.

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis). The coordinates are given as (r, θ), where 'r' is the radial distance and 'θ' is the angle in radians. Understanding how to interpret and plot these coordinates is essential for visualizing points in polar systems.

Angle Representation

In polar coordinates, angles can be represented in multiple ways due to the periodic nature of trigonometric functions. For example, an angle of θ can be expressed as θ + 2kπ, where k is any integer, allowing for equivalent angles. This concept is crucial for finding alternative representations of points, especially when adjusting the angle to fit specific ranges.

Negative Radius

When the radial distance 'r' is negative in polar coordinates, it indicates that the point is located in the opposite direction of the angle θ. This means that instead of moving 'r' units from the origin at angle θ, you move 'r' units in the direction of θ + π. Understanding this concept is vital for correctly interpreting and plotting points with negative radii.