In Exercises 1–10, indicate if the point with the given polar coordinates is represented by A, B, C, or D on the graph. (3, π)

Verified step by step guidance

1. Step 1: Understand Polar Coordinates - Polar coordinates are given in the form (r, θ), where r is the radius (distance from the origin) and θ is the angle measured from the positive x-axis.
2. Step 2: Analyze the Given Coordinates - The given polar coordinates are (3, π). This means the point is 3 units away from the origin, and the angle θ is π radians.
3. Step 3: Convert the Angle - Recall that π radians is equivalent to 180 degrees. This means the point lies on the negative x-axis.
4. Step 4: Determine the Position - Since the radius is 3, the point is 3 units to the left of the origin along the x-axis.
5. Step 5: Match with the Graph - Look at the graph and identify which point (A, B, C, or D) is located 3 units to the left of the origin on the x-axis.

Key Concepts

Polar Coordinates

Polar coordinates represent a point 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 first value indicates the radius (distance from the origin), while the second value indicates the angle in radians. For example, the polar coordinates (3, π) mean a point that is 3 units away from the origin at an angle of π radians.

Graphing Polar Coordinates

To graph polar coordinates, one must convert the polar values into Cartesian coordinates or plot them directly on a polar grid. The radius is measured outward from the origin, and the angle is measured counterclockwise from the positive x-axis. For (3, π), the point lies on the negative x-axis, 3 units away from the origin, which is crucial for identifying its representation on a graph.

Understanding Angles in Radians

In trigonometry, angles can be measured in degrees or radians, with radians being the standard unit in polar coordinates. One full rotation (360 degrees) is equivalent to 2π radians. The angle π radians corresponds to 180 degrees, indicating that the point (3, π) is located directly opposite the positive x-axis, reinforcing the importance of understanding angle measurement when interpreting polar coordinates.