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. (4, π/2)

Verified step by step guidance

1. insert step 1: Understand the given polar coordinates (r, θ) = (4, \frac{\pi}{2}). This means the point is 4 units away from the origin at an angle of \frac{\pi}{2} radians from the positive x-axis.
2. insert step 2: Plot the point (4, \frac{\pi}{2}) on the polar coordinate system. This point lies on the positive y-axis, 4 units above the origin.
3. insert step 3: For part (a), find another representation with r > 0 and 2\pi < θ < 4\pi. Add 2\pi to the angle: θ = \frac{\pi}{2} + 2\pi = \frac{5\pi}{2}. The new coordinates are (4, \frac{5\pi}{2}).
4. insert step 4: For part (b), find another representation with r < 0 and 0 < θ < 2\pi. Use the property that (r, θ) = (-r, θ + \pi). So, (-4, \frac{\pi}{2} + \pi) = (-4, \frac{3\pi}{2}).
5. insert step 5: For part (c), find another representation with r > 0 and -2\pi < θ < 0. Subtract 2\pi from the angle: θ = \frac{\pi}{2} - 2\pi = -\frac{3\pi}{2}. The new coordinates are (4, -\frac{3\pi}{2}).

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

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 + 2πk, where k is any integer. This means that angles can be adjusted by adding or subtracting full rotations (2π) to find equivalent angles, which is crucial for finding alternate representations of points.

Negative Radius in Polar Coordinates

When the radius 'r' is negative in polar coordinates, it indicates that the point is located in the opposite direction of the angle θ. This concept is important for understanding how to find alternate representations of points, as a negative radius effectively reflects the point across the origin, allowing for different angle ranges while maintaining the same location in the plane.