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Chapter 10: Q 62 (page 697)

The hypocycloid is a curve defined by the parametric equations

x(t)=cos3t,y(t)=sin3t,0鈮�t鈮�2蟺

(a) Graph the hypocycloid using a graphing utility.

(b) Find a rectangular equation of the hypocycloid.

Short Answer

Expert verified

(a) The graph of the hypocycloid is,

(b) The rectangular equation of the hypocycloid is,
x23+y23=1

Step by step solution

01

Step 1. Given information

The hypocycloid is a curve defined by the parametric equations

x(t)=cos3t,y(t)=sin3t,0鈮�t鈮�2蟺

02

Part (a) of Step 1. Graph of hypocycloid 

The graph of the hypocycloid defined by the parametric equation x(t)=cos3t,y(t)=sin3t,0鈮�t鈮�2蟺is

03

Part (b) of Step 1. The rectangular equation of the hypocycloid.

The parametric equation of the hypocycloid curve is x(t)=cos3t,y(t)=sin3t,0鈮�t鈮�2蟺

Considering the trigonometric equation

The given values can be transformed as

localid="1648366478382" cost=x13sint=y13cos2t=x23sin2t=y23

Substituting the values of cos2tandsin2tin the trigonometric identity we get,

localid="1648366556197" cos2t+sin2t=1x23+y23=1

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