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Classical Curves

Classical Curves. r= a sin n θ r= a cos n θ. ROSE CURVES. SINE: starts Quadrant I. COSINE: starts x axis. CARDIOD. r= a + a sin n θ r= a + a cos n θ. Sine (x axis) Cosine (y axis). a + a (distance point to point) ±a (intercepts). LIMACON. r= a + b sin n θ r= a + b cos n θ.

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Classical Curves

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  1. Classical Curves

  2. r= a sin nθ r= a cos nθ ROSE CURVES SINE: starts Quadrant I COSINE: starts x axis

  3. CARDIOD r= a + a sin nθ r= a + a cos nθ Sine (x axis) Cosine (y axis) a + a (distance point to point) ±a (intercepts)

  4. LIMACON r= a + b sin nθ r= a + b cos nθ a + b (distance shape) b – a (distance loop) ± a intercepts a < b b < a a > 2b loop dimple convex (no shape) Cosine (x axis) Sine (y axis)

  5. LEMNISCATE r2= a sin 2θ r2= a cos 2θ Cosine (x axis) Sine (diagonal)

  6. SPIRAL OF ARCHIMEDES r = aθ More spiral (coefficient decimal/small) Less spiral (coefficient larger)

  7. DESCRIBE EACH R= 3 + 3cosθ R= -2sin3 θ R= -5 + 3cosθ R2= 4cos2θ

  8. r= 2sin3θ

  9. RECTANGULAR TO POLAR R (x, y) P (r, θ) R= θ = Arctan (y/x) IF X IS (+) θ = Arctan (y/x) + πIF X IS (-)

  10. POLAR TO RECTANGULAR P (r, θ) R (x, y) X = r cos θ Y = r sin θ

  11. EQUATIONS Complete the square- Substitution- Trig identities! Polar form of x2 + y2 = 16 (r2cos2θ) + (r2sin2θ)= 16 r2 (cos2θ+ sin2θ) = 16 r2 (1) = 16 r= 4 Substitute values of x & y Factor out “r2” Trig identity “cos2θ+ sin2θ=1” Simplify Rectangular form of r=-secθ r= 1 cos θ rcosθ=1 x=1 Rewrite “sec” as “1/cos” Cross multiply Substitute with “x”

  12. To Polar: R (-1,1) R (8, 8√3) X2 + (y + 6)2 = 36 To Rectangular: P(4, π/6) P (-2, π/3) r= 3sinθ

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