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, 225°)

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 counterclockwise from the positive x-axis.
2. Identify the given coordinates: \(r = 3\) and \(\theta = 225^\circ\).
3. Convert the angle \(225^\circ\) to radians if needed, or understand its position on the unit circle. Since \(225^\circ\) is in the third quadrant (between \(180^\circ\) and \(270^\circ\)), the point lies diagonally down-left from the origin.
4. Plot the point by moving 3 units from the origin in the direction of \(225^\circ\). This means moving 3 units into the third quadrant along the line that makes a \(225^\circ\) angle with the positive x-axis.
5. Compare the plotted point with points A, B, C, and D on the graph to determine which one matches the location of \((3, 225^\circ)\).

Key Concepts

Polar Coordinates System

Polar coordinates represent points in a plane using a distance from the origin (radius) and an angle measured from the positive x-axis. Each point is given as (r, θ), where r is the radius and θ is the angle in degrees or radians.

Angle Measurement and Quadrants

Angles in polar coordinates are measured counterclockwise from the positive x-axis. Understanding which quadrant an angle lies in helps locate the point; for example, 225° lies in the third quadrant, where both x and y coordinates are negative.

Plotting Points from Polar to Cartesian Coordinates

To plot a polar point on a Cartesian graph, convert (r, θ) to (x, y) using x = r cos θ and y = r sin θ. This conversion helps identify the exact location of the point on the graph and match it to labeled points A, B, C, or D.