Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo

Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problema 27

Capitolo 5, Problema 27

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 4 sin 3θ

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Recall the three common symmetry tests for 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 symmetry about the polar axis, replace \( \theta \) with \( -\theta \) in the equation and check if the equation remains unchanged. For \( r = 4 \sin 3\theta \), substitute \( -\theta \) to get \( r = 4 \sin(-3\theta) \).

To test symmetry about the line \( \theta = \frac{\pi}{2} \), replace \( \theta \) with \( \pi - \theta \) and check if the equation remains unchanged. Substitute \( \pi - \theta \) into \( r = 4 \sin 3\theta \) to get \( r = 4 \sin 3(\pi - \theta) \).

To test symmetry about the pole, replace \( r \) with \( -r \) and \( \theta \) with \( \theta + \pi \) and check if the equation remains unchanged. Substitute these into the equation to get \( -r = 4 \sin 3(\theta + \pi) \).

After determining the symmetries, sketch the graph by plotting points for values of \( \theta \) from 0 to \( 2\pi \), using the equation \( r = 4 \sin 3\theta \). Note that the factor 3 inside the sine function indicates the graph will have multiple petals (specifically, 3 petals if \( n \) is odd in \( r = a \sin n\theta \)).

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Polar Coordinates and Equations

Polar coordinates represent points using a radius and an angle (r, θ) instead of Cartesian coordinates. Understanding how to interpret and plot equations like r = 4 sin 3θ is essential for graphing curves in the polar plane.

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Intro to Polar Coordinates

Symmetry Tests in Polar Graphs

Testing symmetry in polar graphs involves checking if the graph is symmetric about the polar axis, the line θ = π/2, or the pole. This is done by substituting θ with -θ, π - θ, or replacing r with -r, helping to simplify graphing and understand the curve's shape.

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Cardioids

Graphing Rose Curves

Equations of the form r = a sin(nθ) or r = a cos(nθ) produce rose curves with petals. The number of petals depends on n: if n is odd, the curve has n petals; if even, it has 2n petals. Recognizing this helps in sketching the graph of r = 4 sin 3θ accurately.

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Roses

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