Domanda del libro di testo
In Exercises 61–63, test for symmetry with respect to
a. the polar axis.
b. the line θ = π/2.
c. the pole.
r = 5 + 3 cos θ
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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 5.RE.55
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 5.RE.55Capitolo 5, Problema 5.RE.55
In Exercises 54–60, convert each polar equation to a rectangular equation. Then use your knowledge of the rectangular equation to graph the polar equation in a polar coordinate system. θ = 3π/4
Guida verificata passo dopo passo1
Recall the relationship between polar and rectangular coordinates: \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).
Given the polar equation \(\theta = \frac{3\pi}{4}\), recognize that this represents all points where the angle from the positive x-axis is \(\frac{3\pi}{4}\) radians.
Express \(\tan{\theta}\) in terms of \(x\) and \(y\) using the identity \(\tan{\theta} = \frac{y}{x}\).
Substitute \(\theta = \frac{3\pi}{4}\) into the tangent expression to get \(\tan{\frac{3\pi}{4}} = \frac{y}{x}\).
Use the known value of \(\tan{\frac{3\pi}{4}}\) to write the rectangular equation relating \(x\) and \(y\), which can then be used to graph the line in the rectangular coordinate system and interpret it in the polar coordinate system.
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Polar and Rectangular Coordinate Systems
Polar coordinates represent points using a radius and an angle (r, θ), while rectangular coordinates use (x, y) positions on a plane. Understanding how these systems relate is essential for converting equations and interpreting graphs in both formats.
Video consigliato:
Percorso guidato
05:32
Intro to Polar Coordinates
Conversion Formulas Between Polar and Rectangular Coordinates
The key formulas for conversion are x = r cos θ and y = r sin θ, which translate polar coordinates into rectangular form. Additionally, r = √(x² + y²) and θ = arctan(y/x) convert rectangular coordinates back to polar.
Video consigliato:
Percorso guidato
06:17Convert Points from Polar to Rectangular
Graphing Lines in Polar Coordinates
A polar equation like θ = constant represents a line through the origin at a fixed angle. Converting this to rectangular form helps visualize the line as y = mx, where m = tan(θ), facilitating graphing in both coordinate systems.
Video consigliato:
Percorso guidato
05:32Intro to Polar Coordinates
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