Question #145249
Find the evolute of the four cuspecl hypocycloid
x=acos^3θ y=asin^3θ
1
Expert's answer
2020-11-23T08:54:12-0500

The equation of the evolute is



r(t)=x(t)i+y(t)j{X(t)=x(t)y(t)(x(t)2+y(t)2)x(t)y(t)x(t)y(t)Y(t)=y(t)+x(t)(x(t)2+y(t)2)x(t)y(t)x(t)y(t)\vec{r}(t)=x(t)\cdot\vec{i}+y(t)\cdot\vec{j}\to\\[0.3cm] \left\{\begin{array}{l} X(t)=x(t)-\displaystyle\frac{y'(t)\cdot\left(x'(t)^2+y'(t)^2\right)}{x'(t)y''(t)-x''(t)y'(t)}\\[0.5cm] Y(t)=y(t)+\displaystyle\frac{x'(t)\cdot\left(x'(t)^2+y'(t)^2\right)}{x'(t)y''(t)-x''(t)y'(t)} \end{array}\right.

(more infotmation : https://en.wikipedia.org/wiki/Evolute )


In our case,



{x(θ)=acos3(θ)y(θ)=asin3(θ){x(θ)=3acos2(θ)sin(θ)x(θ)=3a2cos(θ)sin(2θ)x(θ)=3a2(sin(θ)sin(2θ)+2cos(θ)cos(2θ))y(θ)=3asin2(θ)cos(θ)y(θ)=3a2sin(θ)sin(2θ)y(θ)=3a2(cos(θ)sin(2θ)+2sin(θ)cos(2θ))\left\{\begin{array}{c} x(\theta)=a\cos^3(\theta)\\[0.3cm] y(\theta)=a\sin^3(\theta) \end{array}\right.\to\\[0.3cm] \left\{\begin{array}{c} x'(\theta)=-3a\cos^2(\theta)\cdot\sin(\theta)\\[0.3cm] \boxed{x'(\theta)=\displaystyle\frac{-3a}{2}\cos(\theta)\sin(2\theta)}\\[0.5cm] x''(\theta)=\displaystyle\frac{-3a}{2}(-\sin(\theta)\sin(2\theta)+2\cos(\theta)\cos(2\theta))\\[0.5cm] y'(\theta)=3a\sin^2(\theta)\cos(\theta)\\[0.3cm] \boxed{y'(\theta)=\displaystyle\frac{3a}{2}\sin(\theta)\sin(2\theta)}\\[0.5cm] y''(\theta)=\displaystyle\frac{3a}{2}(\cos(\theta)\sin(2\theta)+2\sin(\theta)\cos(2\theta)) \end{array}\right.

Now we transform the expressions that enter the equations for the evolute :



x(θ)2+y(θ)2=(3a2cos(θ)sin(2θ))2+(3a2sin(θ)sin(2θ))2=9a24sin2(2θ)(cos2(θ)+sin2(θ))9a24sin2(2θ)x(θ)2+y(θ)2=9a24sin2(2θ)x(θ)y(θ)x(θ)y(θ)==3a2cos(θ)sin(2θ)3a2(cos(θ)sin(2θ)+2sin(θ)cos(2θ))3a2(sin(θ)sin(2θ)+2cos(θ)cos(2θ))3a2sin(θ)sin(2θ)==9a24sin(2θ)(cos2(θ)sin(2θ)+2sin(θ)cos(θ)cos(2θ)++sin2(θ)sin(2θ)2sin(θ)cos(θ)cos(2θ))==9a24sin(2θ)(sin(2θ)(sin2(θ)+cos2(θ)))==9a24sin2(2θ)x(θ)y(θ)x(θ)y(θ)=9a24sin2(2θ)x'(\theta)^2+y'(\theta)^2=\left(\frac{-3a}{2}\cos(\theta)\sin(2\theta)\right)^2+\left(\frac{3a}{2}\sin(\theta)\sin(2\theta)\right)^2\\[0.3cm] =\frac{9a^2}{4}\cdot\sin^2(2\theta)\cdot\left(\cos^2(\theta)+\sin^2(\theta)\right)\equiv\frac{9a^2}{4}\cdot\sin^2(2\theta)\\[0.3cm] \boxed{x'(\theta)^2+y'(\theta)^2=\frac{9a^2}{4}\cdot\sin^2(2\theta)}\\[0.3cm] x'(\theta)y''(\theta)-x''(\theta)y'(\theta)=\\[0.3cm] =\frac{-3a}{2}\cos(\theta)\sin(2\theta)\cdot\frac{3a}{2}(\cos(\theta)\sin(2\theta)+2\sin(\theta)\cos(2\theta))-\\[0.3cm] -\frac{-3a}{2}(-\sin(\theta)\sin(2\theta)+2\cos(\theta)\cos(2\theta))\cdot\frac{3a}{2}\sin(\theta)\sin(2\theta)=\\[0.3cm] =\frac{-9a^2}{4}\sin(2\theta)\cdot\left(\cos^2(\theta)\sin(2\theta)+2\sin(\theta)\cos(\theta)\cos(2\theta)+\right.\\[0.3cm] \left.+\sin^2(\theta)\sin(2\theta)-2\sin(\theta)\cos(\theta)\cos(2\theta)\right)=\\[0.3cm] =\frac{-9a^2}{4}\sin(2\theta)\cdot\left(\sin(2\theta)\cdot\left(\sin^2(\theta)+\cos^2(\theta)\right)\right)=\\[0.3cm] =\frac{-9a^2}{4}\sin^2(2\theta)\\[0.3cm] \boxed{x'(\theta)y''(\theta)-x''(\theta)y'(\theta)=\frac{-9a^2}{4}\sin^2(2\theta)}

Substitute the found expressions into the equations for the evolute, which were indicated at the very beginning :



X(θ)=x(θ)y(θ)(x(θ)2+y(θ)2)x(θ)y(θ)x(θ)y(θ)==acos3(θ)3a2sin(θ)sin(2θ)9a24sin2(2θ)9a24sin2(2θ)==acos3(θ)+3a2sin(θ)sin(2θ)=acos3(θ)+3asin2(θ)cos(θ)==acos(θ)(cos2(θ)+3sin2(θ))acos(θ)(1+2sin2(θ))X(θ)=acos(θ)(1+2sin2(θ))Y(θ)=y(θ)+x(θ)(x(θ)2+y(θ)2)x(θ)y(θ)x(θ)y(θ)==asin3(θ)+3a2cos(θ)sin(2θ)9a24sin2(2θ)9a24sin2(2θ)==asin3(θ)+3a2cos(θ)sin(2θ)=asin3(θ)+3acos2(θ)sin(θ)==asin(θ)(sin2(θ)+3cos2(θ))asin(θ)(1+2cos2(θ))Y(θ)=asin(θ)(1+2cos2(θ))X(\theta)=x(\theta)-\frac{y'(\theta)\cdot\left(x'(\theta)^2+y'(\theta)^2\right)}{x'(\theta)y''(\theta)-x''(\theta)y'(\theta)}=\\[0.3cm] =a\cos^3(\theta)-\frac{\displaystyle\frac{3a}{2}\sin(\theta)\sin(2\theta)\cdot\displaystyle\frac{9a^2}{4}\sin^2(2\theta)}{\displaystyle\frac{-9a^2}{4}\sin^2(2\theta)}=\\[0.3cm] =a\cos^3(\theta)+\frac{3a}{2}\sin(\theta)\sin(2\theta)=a\cos^3(\theta)+3a\sin^2(\theta)\cos(\theta)=\\[0.3cm] =a\cos(\theta)\left(\cos^2(\theta)+3\sin^2(\theta)\right)\equiv a\cos(\theta)\left(1+2\sin^2(\theta)\right)\\[0.3cm] \boxed{X(\theta)=a\cos(\theta)\left(1+2\sin^2(\theta)\right)}\\[0.3cm] Y(\theta)=y(\theta)+\frac{x'(\theta)\cdot\left(x'(\theta)^2+y'(\theta)^2\right)}{x'(\theta)y''(\theta)-x''(\theta)y'(\theta)}=\\[0.3cm] =a\sin^3(\theta)+\frac{\displaystyle\frac{-3a}{2}\cos(\theta)\sin(2\theta)\cdot\displaystyle\frac{9a^2}{4}\sin^2(2\theta)}{\displaystyle\frac{-9a^2}{4}\sin^2(2\theta)}=\\[0.3cm] =a\sin^3(\theta)+\frac{3a}{2}\cos(\theta)\sin(2\theta)=a\sin^3(\theta)+3a\cos^2(\theta)\sin(\theta)=\\[0.3cm] =a\sin(\theta)\left(\sin^2(\theta)+3\cos^2(\theta)\right)\equiv a\sin(\theta)\left(1+2\cos^2(\theta)\right)\\[0.3cm] \boxed{Y(\theta)=a\sin(\theta)\left(1+2\cos^2(\theta)\right)}\\[0.3cm]

ANSWER



{x(θ)=acos3(θ)y(θ)=asin3(θ){X(θ)=acos(θ)(1+2sin2(θ))Y(θ)=asin(θ)(1+2cos2(θ))\left\{\begin{array}{c} x(\theta)=a\cos^3(\theta)\\[0.3cm] y(\theta)=a\sin^3(\theta) \end{array}\right.\to \left\{\begin{array}{c} X(\theta)=a\cos(\theta)\left(1+2\sin^2(\theta)\right)\\[0.3cm] Y(\theta)=a\sin(\theta)\left(1+2\cos^2(\theta)\right) \end{array}\right.


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS