In Exercises 11–20, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. (−1, π)

Verified step by step guidance

1. Understand that polar coordinates are given in the form \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis.
2. Identify the given polar coordinates: \((-1, \pi)\). Here, \(r = -1\) and \(\theta = \pi\).
3. Recognize that a negative radius means the point is in the opposite direction of the angle \(\theta\).
4. Plot the angle \(\theta = \pi\) on the polar coordinate system, which corresponds to the negative x-axis.
5. Since the radius \(r = -1\) is negative, move 1 unit in the opposite direction of \(\pi\), which places the point at \((1, 0)\) in Cartesian coordinates.

Key Concepts

Polar Coordinates

Polar coordinates represent points in a two-dimensional space using a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis). The format is (r, θ), where 'r' is the radial distance and 'θ' is the angle in radians. Understanding this system is crucial for plotting points accurately in polar form.

Negative Radius in Polar Coordinates

In polar coordinates, a negative radius indicates that the point is located in the opposite direction of the angle specified. For example, the point (−1, π) means to move 1 unit in the direction opposite to the angle π (which points left along the negative x-axis), effectively placing the point at (1, 0) in Cartesian coordinates.

Conversion Between Polar and Cartesian Coordinates

Converting between polar and Cartesian coordinates is essential for visualizing points. The formulas are x = r * cos(θ) and y = r * sin(θ) for converting from polar to Cartesian, and r = √(x² + y²) and θ = arctan(y/x) for the reverse. This conversion helps in understanding the location of points in different coordinate systems.