Question

Eliminate $\theta$ from $x=a{\sin }^3\theta ,y=b\cos {}^3\theta$.

Answer 1

Given $x=a{\sin }^3\theta ,y=b\cos {}^3\theta$.

Now from the given equations we get,

$\sin \theta ={\left(\frac{x}{a}\right)}^{\frac{1}{3}}$.......(1) and $\cos \theta ={\left(\frac{y}{b}\right)}^{\frac{1}{3}}$.......(2).

As we've,

${\sin }^2\theta +{\cos }^2\theta =1$ then we get form (1) and (2) we get,

${\left(\frac{x}{a}\right)}^{\frac{2}{3}}+{\left(\frac{y}{b}\right)}^{\frac{2}{3}}=1$.