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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblem 11.4.6
Chapter 11, Problem 11.4.6

Symmetries and Polar Graphs


Identify the symmetries of the curves in Exercises 1–12. Then sketch the curves in the xy-plane.


r = 1 + 2 sin θ

Verified step by step guidance
1
Recall that to identify symmetries of a polar curve given by \(r = f(\theta)\), we check for symmetry about the polar axis (the x-axis), the line \(\theta = \frac{\pi}{2}\) (the y-axis), and the pole (origin).
For symmetry about the polar axis, replace \(\theta\) by \(-\theta\) and see if the equation remains unchanged or can be manipulated to the original form. That is, check if \(r = 1 + 2 \sin(-\theta)\) simplifies to the original \(r\).
For symmetry about the line \(\theta = \frac{\pi}{2}\), replace \(\theta\) by \(\pi - \theta\) and check if \(r = 1 + 2 \sin(\pi - \theta)\) simplifies to the original \(r\).
For symmetry about the pole (origin), replace \(r\) by \(-r\) and \(\theta\) by \(\theta + \pi\), and check if \(-r = 1 + 2 \sin(\theta + \pi)\) can be rearranged to the original equation.
After determining the symmetries, sketch the curve by plotting points for various values of \(\theta\) between \(0\) and \(2\pi\), using the equation \(r = 1 + 2 \sin \theta\), and then reflect the curve according to the symmetries found.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polar Coordinates and Graphing

Polar coordinates represent points using a radius r and an angle θ, differing from Cartesian coordinates. Understanding how to plot points given r(θ) is essential for sketching curves like r = 1 + 2 sin θ, where r changes with θ from 0 to 2π.
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Intro to Polar Coordinates

Symmetry in Polar Graphs

Symmetry in polar graphs can occur about the polar axis, the line θ = π/2, or the pole (origin). Testing the function for transformations like replacing θ with -θ, π - θ, or θ + π helps identify these symmetries, which simplifies sketching and understanding the curve.
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Intro to Polar Coordinates

Trigonometric Functions in Polar Equations

Trigonometric functions like sine and cosine influence the shape and symmetry of polar graphs. For r = 1 + 2 sin θ, the sine term affects the radius depending on θ, creating characteristic shapes such as limacons, cardioids, or loops, which are important to recognize.
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Introduction to Trigonometric Functions
Related Practice
Textbook Question

Implicitly Defined Parametrizations


Assuming that the equations in Exercises 15−20 define x and y implicitly as differentiable functions x=f(t), y=g(t), find the slope of the curve x=f(t), y=g(t) at the given value of t.


x sin t + 2x = t, t sin t − 2t = y, t = π

Textbook Question

Finding Parametric Equations


In Exercises 31–36, find a parametrization for the curve.


the ray (half line) with initial point (-1,2) that passes through the point (0,0)

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Textbook Question

Finding Parametric Equations


In Exercises 31–36, find a parametrization for the curve.


the line segment with endpoints (-1,3) and (3,-2)

Textbook Question

Finding Cartesian from Parametric Equations


Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.


x = 1 + sin t, y = cos t − 2, 0 ≤ t ≤ π

Textbook Question

Finding Polar Areas


Find the areas of the regions in Exercises 9–18.


Inside the circle r = 4 sin θ and below the horizontal line r = 3 csc θ

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Textbook Question

Theory and Examples


Tangents Find equations for the tangents to the circle (x − 2)² + (y − 1)² = 5 at the points where the circle crosses the coordinate axes.