Q&A

# Find $y=b\cos^3\theta$ and, $x=a\sin^3\theta$ from hypocycloid's formula

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Here's the equation for hypocycloid $$(\frac{x}{a})^{\dfrac{2}{3}}+(\frac{y}{b})^{\dfrac{2}{3}}=1$$

Now, I have to find an equation for x and y. I can simply find by simple algebra. But, my book had used parametric equation, and they wrote

$$y=b\cos^3\theta, x=a\sin^3\theta$$

How to find parametric equation? I don't have separated equation for x and y so, I can't do this way

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We know $$\sin^2\theta+\cos^2\theta=1$$ So, we can write $$(\sin^3 \theta)^{\dfrac{2}{3}} +(\cos^3\theta)^{\dfrac{2}{3}}=1$$

Let, $$\sin^3\theta=\frac{x}{a}$$ $$\cos^3\theta=\frac{y}{b}$$

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

So, they took $$x=a\sin^3 \theta$$ and, $$y=a \cos^3 \theta$$

That's what they did

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