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. Recall that polar coordinates are given in the form \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle measured from the positive x-axis (polar axis) in radians.
2. Identify the given coordinates: \(r = 3\) and \(\theta = \frac{4\pi}{3}\) radians. This means the point is 3 units away from the origin at an angle of \(\frac{4\pi}{3}\) radians.
3. Convert the angle \(\frac{4\pi}{3}\) radians to degrees if needed for easier visualization: multiply by \(\frac{180}{\pi}\) to get degrees, but this step is optional if you are comfortable with radians.
4. On the polar coordinate system, start at the positive x-axis and rotate counterclockwise by the angle \(\frac{4\pi}{3}\) radians. This places you in the third quadrant since \(\frac{4\pi}{3}\) is greater than \(\pi\) (which is \(180^\circ\)).
5. From the origin, move outward along the line at angle \(\frac{4\pi}{3}\) by a distance of 3 units. Mark this point on the graph.

Key Concepts

Polar Coordinate System

The polar coordinate system represents points in a plane using a distance from the origin (radius r) and an angle (θ) measured from the positive x-axis. Each point is expressed as (r, θ), where r ≥ 0 and θ is typically in radians or degrees.

Plotting Points Using Polar Coordinates

To plot a point given in polar coordinates (r, θ), start at the origin, rotate counterclockwise by the angle θ, then move outward along that direction by the distance r. Negative values of r indicate moving in the opposite direction of the angle θ.

Angle Measurement in Radians

Angles in polar coordinates are often measured in radians, where 2π radians equal 360 degrees. Understanding radian measure helps accurately locate points on the plane, such as 4π/3 radians, which corresponds to 240 degrees.