circle of radius 1 rolls around the circumference of a larg cle of radius 4 , as shown in Figure 10.33 . The epicycloid ced by a point on the circumference of the smaller circle ven by $$ \quad x=5 \cos t-\cos 5 t $$ and $$ y=5 \sin t-\sin 5 t $$

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We start with the general equation for an epicycloid generated by a circle of radius $$ r $$ rolling externally around a fixed circle of radius $$ R $$. The standard parametric equations are

$$
x = (R + r) \cos t - r \cos\!\left(\frac{R + r}{r}\, t\right)
$$
$$
y = (R + r) \sin t - r \sin\!\left(\frac{R + r}{r}\, t\right)
$$

In our problem the smaller circle has radius $$ r = 1 $$ and the larger circle has radius $$ R = 4 $$. Substituting these values in gives:

$$
x = (4 + 1) \cos t - 1 \cdot \cos\!\left(\frac{4+1}{1}\, t\right) = 5 \cos t - \cos 5t,
$$
$$
y = (4 + 1) \sin t - 1 \cdot \sin\!\left(\frac{4+1}{1}\, t\right) = 5 \sin t - \sin 5t.
$$

Thus, the epicycloid traced by a point on the circumference of the smaller circle is given by

$$
x = 5 \cos t - \cos 5t \quad \text{and} \quad y = 5 \sin t - \sin 5t.
$$
The epicycloid traced by a point on the smaller circle is given by: $$ x = 5 \cos t - \cos 5t \quad \text{and} \quad y = 5 \sin t - \sin 5t. $$

circle of radius 1 rolls around the circumference of a larg cle of radius 4 , as shown in Figure 10.33 . The epicycloid ced by a point on the circumference of the smaller circle ven by \( \quad x=5 \cos t-\cos 5 t \) and \( y=5 \sin t-\sin 5 t \)

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