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°)

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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.

Identify the given coordinates: \(r = 3\) and \(\theta = 225^\circ\).

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.

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.

Compare the plotted point with points A, B, C, and D on the graph to determine which one matches the location of \((3, 225^\circ)\).

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주요 개념

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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.