Classical Curves

ROSE CURVES r= a sin n θ r= a cos n θ SINE: starts Quadrant I COSINE: starts x axis

CARDIOD r= a + a sin n θ r= a + a cos n θ a + a (distance point to point) ±a (intercepts) Sine (x axis) Cosine (y axis)

LIMACON a + b (distance shape) b – a (distance loop) ± a intercepts a 2b loopdimpleconvex (no shape) r= a + b sin n θ r= a + b cos n θ Cosine (x axis) Sine (y axis)

LEMNISCATE r 2 = a sin 2 θ r 2 = a cos 2 θ Cosine (x axis) Sine (diagonal)

SPIRAL OF ARCHIMEDES r = a θ More spiral (coefficient decimal/small) Less spiral (coefficient larger)

DESCRIBE EACH R= 3 + 3cosθR= -2sin3 θ R= cosθR 2 = 4cos2 θ

r= 2sin3θ θ 2sin3 θ 0°0° 15° 30° 45° 60 °

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

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

EQUATIONS Complete the square- Substitution- Trig identities! Polar form of x 2 + y 2 = 16 (r 2 cos 2 θ) + (r 2 sin 2 θ)= 16 r 2 (cos 2 θ+ sin 2 θ) = 16 r 2 (1) = 16 r= 4 Rectangular form of r=-secθ r= 1 cos θ rcos θ=1 x=1 Substitute values of x & y Factor out “r 2 ” Trig identity “ cos 2 θ+ sin 2 θ=1” Simplify Rewrite “sec” as “1/cos” Cross multiply Substitute with “x”

To Polar: R (-1,1) R (8, 8√3) X 2 + (y + 6) 2 = 36 To Rectangular: P(4, π /6) P (-2, π /3) r= 3sin θ