Question

If \[x = a{\sin ^3}\theta \] and \[y = a{\cos ^3}\theta \], then find the value of \[\dfrac{{dy}}{{dx}}\].

Answer 1

Hint: First find the value of \[\dfrac{{dx}}{{d\theta }}\] and \[\dfrac{{dy}}{{d\theta }}\], then divide them to get \[\dfrac{{dy}}{{dx}}\]. Simplify the answer to express it in terms of x and y.

Complete step-by-step answer:

Let us start solving by finding the expression for \[\dfrac{{dx}}{{d\theta }}\].
\[\dfrac{{dx}}{{d\theta }} = \dfrac{d}{{d\theta }}(a{\sin ^3}\theta )\]
We can use \[\dfrac{d}{{dx}}(a{x^3}) = 3a{x^2}\] to simplify the equation.
\[\dfrac{{dx}}{{d\theta }} = 3a{\sin ^2}\theta \dfrac{d}{{d\theta }}(\sin \theta )\]
We know that \[\dfrac{d}{{dx}}(\sin x) = \cos x\], hence, we have the following:
\[\dfrac{{dx}}{{d\theta }} = 3a{\sin ^2}\theta \cos \theta .............(1)\]
Now, let us find the expression for \[\dfrac{{dy}}{{d\theta }}\].
\[\dfrac{{dy}}{{d\theta }} = \dfrac{d}{{d\theta }}(a{\cos ^3}\theta )\]
We can use \[\dfrac{d}{{dx}}(a{x^3}) = 3a{x^2}\] to simplify the equation.
\[\dfrac{{dx}}{{d\theta }} = 3a{\cos ^2}\theta \dfrac{d}{{d\theta }}(\cos \theta )\]
We know that \[\dfrac{d}{{dx}}(\cos x) = - \sin x\], hence, we have the following:
\[\dfrac{{dx}}{{d\theta }} = - 3a{\cos ^2}\theta \sin \theta ............(2)\]
We know that,
\[\dfrac{{dy}}{{dx}} = \dfrac{{\dfrac{{dy}}{{d\theta }}}}{{\dfrac{{dx}}{{d\theta }}}}...........(3)\]
Using equation (1) and equation (2) in equation (3), we have:

\[\dfrac{{dy}}{{dx}} = \dfrac{{3a{{\sin }^2}\theta \cos \theta }}{{ - 3a{{\cos }^2}\theta \sin \theta }}\]
Cancelling common terms in the numerator and the denominator we have:
\[\dfrac{{dy}}{{dx}} = - \dfrac{{\sin \theta }}{{\cos \theta }}\]
We know that \[\dfrac{{\sin \theta }}{{\cos \theta }} = \tan \theta \], hence we have:
\[\dfrac{{dy}}{{dx}} = - \tan \theta ..........(4)\]
We can write equation (4) in terms of x and y.
Let us find the value of \[\dfrac{x}{y}\].
\[\dfrac{x}{y} = \dfrac{{a{{\sin }^3}\theta }}{{a{{\cos }^3}\theta }}\]
Simplifying, we get:
\[\dfrac{x}{y} = \dfrac{{{{\sin }^3}\theta }}{{{{\cos }^3}\theta }}\]
\[\dfrac{x}{y} = {\tan ^3}\theta \]
Let us compute \[\tan \theta \] in terms of x and y by taking the cube root on both sides.
\[\tan \theta = \sqrt[3]{{\dfrac{x}{y}}}.........(5)\]
Substituting equation (5) in equation (4), we get:
\[\dfrac{{dy}}{{dx}} = - \sqrt[3]{{\dfrac{x}{y}}}\]
Hence, the answer is \[ - \sqrt[3]{{\dfrac{x}{y}}}\].

Note: If you express the final answer in terms of \[\theta \], it is a wrong answer. Express the final answer in terms of x and y only.