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, 5π/4)

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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.

Note that the given point is \((-3, \frac{5\pi}{4})\). Since \(r\) is negative, the point lies in the direction opposite to the angle \(\frac{5\pi}{4}\).

Find the angle directly opposite to \(\frac{5\pi}{4}\) by adding or subtracting \(\pi\) (180 degrees): calculate \(\theta_{opposite} = \frac{5\pi}{4} - \pi = \frac{\pi}{4}\).

Plot the point by moving a distance of \(3\) units (the absolute value of \(r\)) from the origin in the direction of \(\theta_{opposite} = \frac{\pi}{4}\).

Compare this location with points A, B, C, and D on the graph to determine which one corresponds to the coordinates \((-3, \frac{5\pi}{4})\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polar Coordinates

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

Negative Radius in Polar Coordinates

A negative radius means the point is located in the direction opposite to the angle θ. To plot (−r, θ), you move r units in the direction θ + π (180 degrees), effectively reversing the direction.

Angle Measurement in Radians

Angles in polar coordinates are often given in radians, where 2π radians equal 360 degrees. Understanding how to convert and interpret angles like 5π/4 helps locate the correct direction on the graph.

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