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In Exercises 11–26, plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. −4i
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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 15
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 15Capitolo 5, Problema 15
In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 − sin θ
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Recall the three common tests for symmetry in polar equations: symmetry about the polar axis (x-axis), symmetry about the line \( \theta = \frac{\pi}{2} \) (y-axis), and symmetry about the pole (origin).
To test for symmetry about the polar axis, replace \( \theta \) with \( -\theta \) in the equation \( r = 1 - \sin \theta \) and check if the equation remains unchanged.
To test for symmetry about the line \( \theta = \frac{\pi}{2} \), replace \( \theta \) with \( \pi - \theta \) and check if the equation remains unchanged.
To test for symmetry about the pole, replace \( r \) with \( -r \) and \( \theta \) with \( \theta + \pi \) and check if the equation remains unchanged.
After determining the symmetries, plot points for various values of \( \theta \) (for example, \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{2}, \pi \), etc.) by calculating corresponding \( r \) values using \( r = 1 - \sin \theta \), then sketch the graph accordingly.
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Polar Coordinates and Equations
Polar coordinates represent points using a radius r and an angle θ from the positive x-axis. Polar equations express r as a function of θ, describing curves in the plane. Understanding how to interpret and plot these equations is essential for graphing polar curves.
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Percorso guidato
05:32
Intro to Polar Coordinates
Symmetry Tests in Polar Graphs
Symmetry in polar graphs can be tested about the polar axis, the line θ = π/2, and the pole (origin). These tests involve substituting θ with -θ, π - θ, or replacing r with -r to check if the equation remains unchanged, helping to simplify graphing and understand the curve's shape.
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Percorso guidato
3:19Cardioids
Graphing Polar Equations Involving Sine
Polar equations with sine functions, like r = 1 − sin θ, often produce limaçon or cardioid shapes. Recognizing the effect of the sine term on the radius as θ varies helps in sketching the curve accurately, noting key points such as maximum and minimum r values.
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Percorso guidato
3:47Introduction to Common Polar Equations
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