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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- 0. Review of College Algebra4h 45m
- 1. Measuring Angles40m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
9. Polar Equations
Polar Coordinate System
Problem 3
Textbook Question
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)
Verified step by step guidance1
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.
Recommended video:
Intro to Polar Coordinates
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.
Recommended video:
Intro to Polar Coordinates
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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Converting between Degrees & Radians
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