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Tangent line at the origin Find the polar equation of the line tangent to the polar curve r=4cosθ at the origin. Explain why the slope of this line is undefined.
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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 12.2.6
What is the polar equation of the horizontal line y = 5?
Guida verificata passo dopo passo1
Recall the relationship between Cartesian coordinates \((x, y)\) and polar coordinates \((r, \theta)\): \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).
Since the line is given by \(y = 5\), substitute \(y\) with \(r \sin{\theta}\) to get the equation \(r \sin{\theta} = 5\).
To express \(r\) in terms of \(\theta\), solve the equation for \(r\): \(r = \frac{5}{\sin{\theta}}\).
Note that this equation is valid for values of \(\theta\) where \(\sin{\theta} \neq 0\), which corresponds to angles where the line \(y=5\) exists in polar coordinates.
Thus, the polar equation of the horizontal line \(y=5\) is \(r = \frac{5}{\sin{\theta}}\).
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Polar Coordinates
Polar coordinates represent points in the plane using a radius and an angle, denoted as (r, θ), where r is the distance from the origin and θ is the angle measured from the positive x-axis. This system is useful for describing curves that are difficult to express in Cartesian coordinates.
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05:32
Intro to Polar Coordinates
Conversion Between Cartesian and Polar Coordinates
To convert between Cartesian (x, y) and polar (r, θ) coordinates, use the formulas x = r cos θ and y = r sin θ. Conversely, r = √(x² + y²) and θ = arctan(y/x). These relationships allow expressing Cartesian equations in terms of r and θ.
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05:32Intro to Polar Coordinates
Equation of a Horizontal Line in Polar Form
A horizontal line y = c can be expressed in polar coordinates by substituting y = r sin θ. Thus, the equation becomes r sin θ = c. This form relates the radius and angle for all points on the horizontal line.
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Percorso guidato
05:14Equations of Tangent Lines
Pratica correlata
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39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin.
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15–30. Working with parametric equations Consider the following parametric equations.
a. Eliminate the parameter to obtain an equation in x and y.
b. Describe the curve and indicate the positive orientation.
x = r − 1, y = r³; −4 ≤ r ≤ 4
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Domanda del libro di testo
33–40. Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region.
The region inside the curve r = √(cos θ)
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Domanda del libro di testo
37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.
r = 6 cos θ + 8 sin θ
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Domanda del libro di testo
Spiral arc length Consider the spiral r=4θ, for θ≥0.
a. Use a trigonometric substitution to find the length of the spiral, for 0≤θ≤√8.
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