Textbook Question
Subtle symmetry Without using a graphing utility, determine the symmetries (if any) of the curve r=4-sin (θ/2)
Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 12, Problem 12.2.35
31–36. Converting coordinates Express the following Cartesian coordinates in polar coordinates in at least two different ways.
(-4, 4√3)
Verified step by step guidance1
Recall the formulas to convert Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\):
\(r = \sqrt{x^2 + y^2}\) and \(\theta = \arctan\left(\frac{y}{x}\right)\), but be careful with the quadrant of the point when determining \(\theta\).
Calculate the radius \(r\) using the given coordinates \(x = -4\) and \(y = 4\sqrt{3}\):
\(r = \sqrt{(-4)^2 + (4\sqrt{3})^2}\).
Find the angle \(\theta\) by computing \(\arctan\left(\frac{y}{x}\right) = \arctan\left(\frac{4\sqrt{3}}{-4}\right)\), then adjust \(\theta\) to the correct quadrant since \(x\) is negative and \(y\) is positive (which places the point in the second quadrant).
Express the polar coordinates as \((r, \theta)\) where \(r\) is the radius found in step 2 and \(\theta\) is the angle found in step 3, typically in radians.
For the second method, use the fact that \(\theta\) can also be expressed as \(\pi - |\arctan(\frac{y}{|x|})|\) to directly find the angle in the second quadrant, then write the polar coordinates again as \((r, \theta)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cartesian and Polar Coordinate Systems
Cartesian coordinates represent points using (x, y) values on perpendicular axes, while polar coordinates describe points by a distance from the origin (r) and an angle (θ) from the positive x-axis. Understanding both systems is essential for converting between them.
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Intro to Polar Coordinates
Conversion Formulas Between Cartesian and Polar Coordinates
To convert from Cartesian (x, y) to polar (r, θ), use r = √(x² + y²) to find the radius and θ = arctan(y/x) to find the angle. Adjustments to θ may be needed depending on the quadrant where the point lies.
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05:32Intro to Polar Coordinates
Multiple Representations of Polar Coordinates
Polar coordinates are not unique; the same point can be represented by different pairs (r, θ) by adding or subtracting full rotations (2π) to the angle or by using negative radius values with adjusted angles. Recognizing these variations helps express coordinates in multiple valid ways.
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05:32Intro to Polar Coordinates
Related Practice
Textbook Question
49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.
y = 3
Textbook Question
81–88. Arc length Find the arc length of the following curves on the given interval.
x = 3 cos t, y = 3 sin t + 1; 0 ≤ t ≤ 2π
Textbook Question
81–88. Arc length Find the arc length of the following curves on the given interval.
x = eᵗ sin t, y = eᵗ cos t; 0 ≤ t ≤ 2π
Textbook Question
37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.
r = 6 cos θ + 8 sin θ
Textbook Question
13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.
x² + y²/9 = 1