Plot the point on the polar coordinate system.
Table of contents
- 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 1
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, 225°)
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.
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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Intro to Polar Coordinates
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.
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Quadratic Formula
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.
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Convert Points from Polar to Rectangular
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