Domanda del libro di testo
45–60. Areas of regions Find the area of the following regions.
The region common to the circles r = 2 sin θ and r = 1
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Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 12, Problema 12.R.43
42–43. Intersection points Find the intersection points of the following curves.
r= √(cos3t) and r= √(sin3t)
Guida verificata passo dopo passo1
Rewrite the given curves in terms of their polar coordinates: the first curve is \(r = \sqrt{\cos(3t)}\) and the second curve is \(r = \sqrt{\sin(3t)}\).
To find the intersection points, set the two expressions for \(r\) equal to each other: \(\sqrt{\cos(3t)} = \sqrt{\sin(3t)}\).
Square both sides to eliminate the square roots, giving \(\cos(3t) = \sin(3t)\).
Use the trigonometric identity or rewrite the equation to find values of \(t\) that satisfy \(\cos(3t) = \sin(3t)\), for example by dividing both sides by \(\cos(3t)\) (where defined) to get \(1 = \tan(3t)\), so \(\tan(3t) = 1\).
Solve for \(t\) by finding all angles where \(\tan(3t) = 1\), then substitute these \(t\) values back into either \(r = \sqrt{\cos(3t)}\) or \(r = \sqrt{\sin(3t)}\) to find the corresponding \(r\) values, which together give the intersection points in polar coordinates.
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Polar Coordinates and Parametric Curves
Polar coordinates represent points using a radius and angle (r, t), where r is the distance from the origin and t is the angle. Understanding how curves are defined in polar form, especially with trigonometric functions of multiples of t, is essential to analyze and find intersections.
Video consigliato:
05:32
Intro to Polar Coordinates
Solving Equations Involving Trigonometric Functions
Finding intersection points requires solving equations where expressions involving trigonometric functions like cos(3t) and sin(3t) are equal. Knowledge of trigonometric identities and properties helps simplify and solve these equations for t.
Video consigliato:
Percorso guidato
6:04Introduction to Trigonometric Functions
Conditions for Intersection in Polar Coordinates
Two polar curves intersect where their radius values are equal for the same angle or where their points coincide in the plane. This involves equating r-values and considering the domain restrictions (e.g., square roots require non-negative arguments) to find valid intersection points.
Video consigliato:
05:32Intro to Polar Coordinates
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65–68. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Graph the curve, labeling vertices, foci, asymptotes (if they exist), and directrices.
A hyperbola with vertices (0, ±2) and directrices y = ±1
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53–57. Conic sections
c. Find the eccentricity of the curve.
x²/4 + y²/25 = 1
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Domanda del libro di testo
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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Domanda del libro di testo
Parabola-hyperbola tangency: Let P be the parabola y = px² and H be the right half of the hyperbola x² - y² = 1.
b. At what point does the tangency occur?
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Domanda del libro di testo
Jake’s response Jake responds to Liz (Exercise 33) with a graph that shows his love for her is infinite. Sketch each of the following curves. Which one should Jake send to Liz to get an infinity symbol?
b. r=(½)+sinθ
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