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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblema 11.7.48
Capitolo 11, Problema 11.7.48

Lines


Sketch the lines in Exercises 45–48 and find Cartesian equations for them.


r cos (θ + π/3) = 2

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1
Recognize that the given equation is in polar form: \(r \cos(\theta + \frac{\pi}{3}) = 2\). Our goal is to convert this into a Cartesian equation involving \(x\) and \(y\).
Recall the relationships between polar and Cartesian coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, note that \(r = \sqrt{x^2 + y^2}\) and \(\theta = \arctan(\frac{y}{x})\).
Use the cosine addition formula to expand \(\cos(\theta + \frac{\pi}{3})\): \(\cos(\theta + \frac{\pi}{3}) = \cos \theta \cos \frac{\pi}{3} - \sin \theta \sin \frac{\pi}{3}\).
Substitute the expanded form back into the original equation: \(r (\cos \theta \cos \frac{\pi}{3} - \sin \theta \sin \frac{\pi}{3}) = 2\). Then replace \(r \cos \theta\) with \(x\) and \(r \sin \theta\) with \(y\) to get: \(x \cos \frac{\pi}{3} - y \sin \frac{\pi}{3} = 2\).
Evaluate the constants \(\cos \frac{\pi}{3}\) and \(\sin \frac{\pi}{3}\), then write the resulting linear equation in \(x\) and \(y\). This is the Cartesian equation of the line. Finally, sketch the line using this Cartesian form.

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Polar Coordinates

Polar coordinates represent points in the plane using a radius and an angle, denoted as (r, θ). Here, r is the distance from the origin, and θ is the angle measured from the positive x-axis. Understanding how to interpret and manipulate these coordinates is essential for converting between polar and Cartesian forms.
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Intro to Polar Coordinates

Trigonometric Identities

Trigonometric identities, such as the angle addition formulas, allow simplification of expressions involving angles. For example, cos(θ + π/3) can be expanded using cos(A + B) = cos A cos B - sin A sin B. This is crucial for rewriting the given polar equation into a more workable form.
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Verifying Trig Equations as Identities

Conversion Between Polar and Cartesian Equations

Converting polar equations to Cartesian form involves using the relationships x = r cos θ and y = r sin θ. By expressing r cos(θ + π/3) in terms of x and y, one can derive a Cartesian equation representing the same line, facilitating sketching and further analysis.
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Convert Equations from Polar to Rectangular
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Graphing Conic Sections


Exercises 63-68 give equations for conic sections and tell how many units up or down and to the right or left each curve is to be shifted. Find an equation for the new conic section, and find the new foci, vertices, centers, and asymptotes, as appropriate. If the curve is a parabola, find the new directrix as well.


x²/169 + y²/144 = 1, right 5, up 12

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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=√(t+1), y=√t, t ≥ 0

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Hyperbolas and Eccentricity


In Exercises 17-24, find the eccentricity of the hyperbola. Then find and graph the hyperbola's foci and directrices.


y² − x² = 4

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Hyperbolas and Eccentricity


Exercises 25–28 give the eccentricities and the vertices or foci of hyperbolas centered at the origin of the xy-plane. In each case, find the hyperbola’s standard-form equation in Cartesian coordinates.


Eccentricity: 1.25

Foci: (0, ±5)

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Graphing Conic Sections


Find the eccentricities of the ellipses and hyperbolas in Exercises 59–62. Sketch each conic section. Include the foci, vertices, and asymptotes (as appropriate) in your sketch.


5y² − 4x² = 20

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Polar Coordinates


Exercises 19–22 give the eccentricities of conic sections with one focus at the origin of the polar coordinate plane, along with the directrix for that focus. Find a polar equation for each conic section.


e = 1/3, r sin θ = −6

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