parametric equations with cubed sin and cos

Question

It has been a while since I have had calc 3,

I know how to find the rectangular equation from parametric equations;

however, I do not remember how to find the rectangular equation

given these parametric equations: $x=\cos^3(\theta)$ and $y=\sin^3(\theta).$

Answer 1

Given $$\displaystyle x=\cos^3(\theta)\Rightarrow x^{\frac{1}{3}} = \cos \theta\Rightarrow x^{\frac{2}{3}} = \cos^2 \theta$$

and $$\displaystyle y=\sin^3(\theta)\Rightarrow x^{\frac{1}{3}} = \sin \theta\Rightarrow y^{\frac{2}{3}} = \sin^2 \theta$$

Now Add above two equations, We get

$$\displaystyle x^{\frac{2}{3}}+y^{\frac{2}{3}} = \cos^2 \theta+\sin^2 \theta = 1\Rightarrow x^{\frac{2}{3}}+y^{\frac{2}{3}} = 1$$

Answer 2

Two thirds of a cube in exponent is a square , so comes the Astroid $ x^{\frac{2}{3}} + y^{\frac{2}{3}} =1. $