Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.R.1c

Chapter 12, Problem 12.R.1c

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. The polar coordinates (3, -3π/4) and (-3, π/4) describe the same point in the plane.

Verified step by step guidance

Recall that polar coordinates are given as \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis.

Understand that the point described by polar coordinates \((r, \theta)\) can also be represented by \((-r, \theta + \pi)\) because moving in the opposite direction by \(\pi\) radians with a negative radius points to the same location.

Check the given coordinates: \((3, -\frac{3\pi}{4})\) and \((-3, \frac{\pi}{4})\). Notice that the second point has a negative radius and an angle \(\frac{\pi}{4}\).

Add \(\pi\) to the angle of the second point to see if it matches the first point's angle: \(\frac{\pi}{4} + \pi = \frac{5\pi}{4}\). Since \(-\frac{3\pi}{4}\) is coterminal with \(\frac{5\pi}{4}\) (they differ by \(2\pi\)), the angles correspond to the same direction.

Conclude that because \((-3, \frac{\pi}{4})\) is equivalent to \((3, \frac{5\pi}{4})\), which is coterminal with \((3, -\frac{3\pi}{4})\), both coordinates describe the same point in the plane.

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Key Concepts

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

Polar Coordinates and Their Representation

Polar coordinates represent points in the plane using a radius and an angle, written as (r, θ). The radius r is the distance from the origin, and θ is the angle measured from the positive x-axis. Points can have multiple polar representations due to angle periodicity and sign changes in r.

Equivalence of Polar Coordinates with Negative Radius

A point with coordinates (r, θ) is equivalent to (-r, θ + π) because a negative radius means moving in the opposite direction along the line defined by θ. This property allows different pairs of (r, θ) to represent the same point by adjusting the angle by π when the radius is negative.

Angle Periodicity in Polar Coordinates

Angles in polar coordinates are periodic with period 2π, meaning θ and θ + 2kπ (for any integer k) represent the same direction. This periodicity allows angles outside the standard interval [0, 2π) or (-π, π] to still describe the same point, which is important when comparing coordinates with angles like -3π/4 and π/4.

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Related Practice

Textbook Question

General equations for a circle Prove that the equations

X = a cos t + b sin t, y = c cos t + d sin t

where a, b, c, and d are real numbers, describe a circle of radius R provided a² +c² =b² +d² = R² and ab+cd=0.

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Textbook Question

53–57. Conic sections

d. Make an accurate graph of the curve.

x = 16y²

Textbook Question

53–57. Conic sections

c. Find the eccentricity of the curve.

x²/4 + y²/25 = 1

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Textbook Question

3–6. Eliminating the parameter Eliminate the parameter to find a description of the following curves in terms of x and y. Give a geometric description and the positive orientation of the curve.

x = sin t - 3, y = cos t + 6; 0 ≤ t ≤ π

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Textbook Question

53–57. Conic sections

a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.

x²/4 + y²/25 = 1

Textbook Question

7–8. Parametric curves and tangent lines

a. Eliminate the parameter to obtain an equation in x and y.

x = 8cos t + 1, y = 8sin t + 2, for 0 ≤ t ≤ 2π; t = π/3

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