Eliminate the parameter and graph the plane curve represented by the parametric equations. Use arrows to show the orientation of each plane curve. x = √t , y = t + 1; −∞ < t < ∞
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 5
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, π)
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 from the positive x-axis (polar axis) in radians.
Identify the given polar coordinates: here, \(r = 3\) and \(\theta = \pi\) radians. This means the point is 3 units away from the origin, along the direction of the angle \(\pi\).
Understand that \(\pi\) radians corresponds to an angle of 180 degrees, which points directly to the left along the negative x-axis.
To locate the point on the graph, start at the origin, move along the angle \(\pi\) (to the left), and measure a distance of 3 units from the origin in that direction.
Compare this position with the points labeled A, B, C, and D on the graph to determine which one matches the coordinates \((3, \pi)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Coordinates System
The polar coordinate system represents points using a distance from the origin (radius r) and an angle θ measured from the positive x-axis. Each point is given as (r, θ), where r ≥ 0 and θ is typically in radians. Understanding how to interpret these coordinates is essential for locating points on a polar graph.
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Intro to Polar Coordinates
Conversion Between Polar and Cartesian Coordinates
To plot or identify points on a graph, it is often helpful to convert polar coordinates (r, θ) into Cartesian coordinates (x, y) using x = r cos θ and y = r sin θ. This conversion allows for easier comparison with points labeled on a Cartesian plane or graph.
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Intro to Polar Coordinates
Angle Measurement and Direction in Polar Coordinates
Angles in polar coordinates are measured counterclockwise from the positive x-axis. Knowing how to interpret the angle θ, especially when it equals π (180 degrees), helps determine the direction of the point from the origin, which is crucial for correctly identifying the point on the graph.
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Intro to Polar Coordinates
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