Domanda del libro di testo
45–60. Areas of regions Find the area of the following regions.
The region outside the circle r = 1/2 and inside the circle r = cos θ
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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 12.2.46
37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.
r = sin θ sec² θ
Guida verificata passo dopo passo1
Recall the relationships between polar and Cartesian coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, \(r^2 = x^2 + y^2\).
Start with the given equation: \(r = \sin \theta \sec^2 \theta\). Rewrite \(\sec^2 \theta\) as \(\frac{1}{\cos^2 \theta}\), so the equation becomes \(r = \sin \theta \cdot \frac{1}{\cos^2 \theta}\).
Multiply both sides of the equation by \(\cos^2 \theta\) to get \(r \cos^2 \theta = \sin \theta\).
Express \(r \cos^2 \theta\) in terms of \(x\) and \(r\): since \(x = r \cos \theta\), then \(r \cos^2 \theta = x \cos \theta\). Also, express \(\sin \theta\) as \(\frac{y}{r}\).
Substitute these into the equation to get \(x \cos \theta = \frac{y}{r}\). Then, express \(\cos \theta\) as \(\frac{x}{r}\) and substitute back to write the entire equation in terms of \(x\), \(y\), and \(r\). Finally, replace \(r\) with \(\sqrt{x^2 + y^2}\) to obtain the Cartesian form of the curve.
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Polar and Cartesian Coordinate Systems
Polar coordinates represent points using a radius and an angle (r, θ), while Cartesian coordinates use x and y values. Understanding how to convert between these systems is essential, using the relationships x = r cos θ and y = r sin θ.
Video consigliato:
05:32
Intro to Polar Coordinates
Trigonometric Identities
Trigonometric identities like sec θ = 1/cos θ and relationships between sine, cosine, and tangent functions help simplify expressions during conversion. Applying these identities allows rewriting polar equations in terms of x and y.
Video consigliato:
7:17Verifying Trig Equations as Identities
Equation Conversion and Curve Description
After substituting polar expressions with Cartesian variables, the resulting equation describes a curve in the xy-plane. Recognizing the form of this equation helps identify the type of curve, such as lines, circles, or conic sections.
Video consigliato:
04:47Introduction to Parametric Equations
Pratica correlata
Domanda del libro di testo
29–32. Intersection points Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.
r = 1 and r = 2 sin 2θ
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Domanda del libro di testo
63–74. Arc length of polar curves Find the length of the following polar curves.
The spiral r = θ², for 0 ≤ θ ≤ 2π
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Domanda del libro di testo
39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin.
A hyperbola with vertices (±2, 0) and asymptotes y = ±3x/2
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Domanda del libro di testo
23–24. Radar Airplanes are equipped with transponders that allow air traffic controllers to see their locations on radar screens. Radar gives the distance of the plane from the radar station (located at the origin) and the angular position of the plane, typically measured in degrees clockwise from north.
A plane is 50 miles from a radar station at an angle of 10 dgeree clockwise from north. Find polar coordinates for the location of the plane.
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Domanda del libro di testo
9–13. Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates.
(-1, -π/3)
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