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.
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.
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.