Question

Find an equation in rectangular coordinates

Answer 1

\[r=4\cos12\theta=4(\cos^34\theta-3\cos4{\theta}\sin^24\theta)\]=

\[r=4\cos4\theta(4\cos^24\theta-3\sin^24\theta)=
4\cos4\theta(4\cos^24\theta-3(1-\cos^24\theta))=4\cos4\theta(7\cos^24\theta-3)\]

\[\cos4\theta=\cos^22\theta-\sin^22\theta\]=\[\cos^2{\theta}-\sin^2\theta-2\sin{\theta}\cos{\theta}\]=\[\frac{ x^2 }{ r^2}-\frac{ y^2 }{ r^2}-2\frac{ xy }{ r^2 }\]

\[r=28\cos^34\theta-12\cos4\theta\]
=\[28(\frac{ x^2 }{ r^2 }-\frac{ y^2 }{ r^2 }-2\frac{ xy }{ r^2 })^3-12(\frac{ x^2 }{ r^2 }-\frac{ y^2 }{ r^2 }-2\frac{ xy }{ r^2 })\]

Answer 2

\[\sin4\theta=2\sin{2\theta}\cos{2\theta}=2(2\sin{\theta}\cos{\theta})(\cos^2{\theta}-\sin^2{\theta})\]=\[4\frac{ xy }{ r^2 }(\frac{ x^2 }{ r^2 }-\frac{ y^2 }{ r^2 })=4(\frac{ x^3y }{ r^4 }-\frac{ xy^3 }{ r^4 })\]

Answer 3

Just in case I need

Answer 4

i'm a bit confused on what you did there, it would be nice if you try to explain it, Thanks