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.(3, 4π/3)

Verified step by step guidance

1. Understand that polar coordinates are given in the form \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle from the positive x-axis.
2. Identify the given polar coordinates: \((3, \frac{4\pi}{3})\). Here, \(r = 3\) and \(\theta = \frac{4\pi}{3}\).
3. Locate the angle \(\theta = \frac{4\pi}{3}\) on the polar coordinate system. This angle is in the third quadrant, as it is more than \(\pi\) but less than \(2\pi\).
4. From the origin, measure a distance of \(r = 3\) units along the direction of the angle \(\frac{4\pi}{3}\).
5. Plot the point at the intersection of the radial line at \(\theta = \frac{4\pi}{3}\) and the circle with radius \(r = 3\).

Key Concepts

Polar Coordinates

Polar coordinates represent points in a plane using a distance from a reference point (the pole) 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. This system is particularly useful for circular and rotational problems.

Plotting Points in Polar Coordinates

To plot a point given in polar coordinates, first identify the radial distance 'r' and the angle 'θ'. The angle is measured counterclockwise from the positive x-axis. For example, for (3, 4π/3), you would move 3 units away from the origin at an angle of 240 degrees (or 4π/3 radians), which places the point in the third quadrant.

Conversion Between Polar and Cartesian Coordinates

Understanding how to convert between polar and Cartesian coordinates is essential. The conversion formulas are x = r * cos(θ) and y = r * sin(θ). This allows you to translate polar coordinates into the familiar (x, y) format, facilitating easier plotting and analysis in a Cartesian plane.