Cardioid
Equation of the chord of the circle with midpoint (1, 0) and radius 3

For the equation of the secant line passing the two points (1 + 3 cos θ, 3 sin θ), (1 + 3 cos 2 θ, 3 sin 2 θ)) {\displaystyle (1+3\cos \theta,3\sin \theta),\ (1+3\cos {\color {red}2}\theta,3\sin {\color {red}2}\theta))} one gets: (sin θ − sin 2 θ) x + (cos 2 θ − sin θ) y = − 2 cos θ − sin (2 θ). {\displaystyle (\sin \theta -\sin 2\theta)x+(\cos 2\theta -\sin \theta)y=-2\cos \theta -\sin(2\theta)\,.}

With help of trigonometric formulae and the subsequent division by sin 1 2 θ {\textstyle \sin {\frac {1}{2}}\theta } the equation of the secant line can be rewritten by: cos (3 2 θ) ⋅ x + sin (3 2 θ) ⋅ y = 4 (cos 1 2 θ) 3 0 < θ < 2 π. {\displaystyle \cos \left({\tfrac {3}{2}}\theta \right)\cdot x+\sin \left({\tfrac {3}{2}}\theta \right)\cdot y=4\left(\cos {\tfrac {1}{2}}\theta \right)^{3}\quad 0<\theta <2\pi.}
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