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Cardioids and Rose Curves

Cardioids and Rose Curves. After Exploration (p. 574). THEOREM: Let a be a (+) real number, then. r = 2a sin  circle: radius a; center (0, a). r = -2a sin  circle: radius a; center (0, -a). r = 2a cos  circle: radius a; center (a, 0).

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Cardioids and Rose Curves

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  1. Cardioids and Rose Curves

  2. After Exploration (p. 574) THEOREM:Let a be a (+) real number, then • r = 2a sin circle: radius a; center (0, a) • r = -2a sin circle: radius a; center (0, -a) • r = 2a cos circle: radius a; center (a, 0) • r = -2a cos circle: radius a; center (-a, 0)

  3. Symmetry I. II. III. (r, ) (r, ) (r, ) (r,   ) (-r, ) (r, -) • symmetric WRT the polar axis (and x-axis) • symmetric WRT  = /2 (and y-axis) • symmetric WRT the pole (and origin) • replace  w/ -. If an equivalent equation results, the graph is symmetric WRT the polar axis • replace  w/   . If an equivalent equation results, the graph is symmetric WRT  = /2 • replace r w/ -r. If an equivalent equation results, the graph is symmetric WRT the pole

  4. Cardioid ~ A heart-shaped curve Forms: r = a(1 + cos) r = a(1  cos) where a > 0, the graph of a cardioid passes through the pole. r = a(1 + sin) r = a(1  sin)

  5. Example: Sketch r = 3(1 + sin)  3(1 + sin) 0 3 30 4.5 60 5.6 90 6 120 5.6 150 4.5 180 3 210 1.5 240 .4 270 0 ** Each circle represents 2 300 .4 330 1.5

  6. The Rose Curve Forms: r = a sin n r = a cos n if n is odd, there are n leaves. if n is even, there are 2n leaves.

  7. Example: Sketch r = sin2 “4 leaf Rose”  sin 2 0 0 30 .866 45 1 60 .866 90 0 Falls on (0,0) 120 -.866 135 -1 150 -.866 Falls on (0,0) 180 0 210 .866 225 1 240 .866 Falls on (0,0) 270 0 300 -.866 315 -1 330 -.866 Falls on (0,0) 360 0 ** Each circle represents .5

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