Parametric functions to make sine curve follow a semicircle of radius $1.5$

Question

Hi I’m supposed to find the parametric formulas to make a sine wave follow a semicircle of radius $1.5$.The curve has to make $10$ periodic oscillations (cycles) moving counterclockwise around the semicircle with the oscillations amplitude of $±0.5$ as shown in the picture:

My answer is that $x(u)= (1.5 + 0.5\sin(20πu))\cos(π)$ and $y(u)= (1.5 + 0.5\sin 20πu))\sin(π)$ but I’m unsure if my approach is correct. Can anyone enlighten me if my answers are correct and if not, how do I go about to solve it?

Answer 1

Your approach is correct. However :

1. $\cos(\pi), \sin(\pi)$ should be $\cos(u), \sin(u)$.

2. The oscillation radius should be $1.5+0.5\sin(20u)$

Answer 2

It is close.

I would start in polar coordinates.

$r = 1.5 + 0.5 \sin 20\theta$

With $0 \le \theta \le \pi$

or

$r = 1.5 + 0.5 \sin 20\pi\theta$

With $0 \le \theta \le 1$

Translate to Cartesian,

$x = r\cos\pi\theta = 1.5\cos\pi\theta + 0.5\cos\theta\sin 20\pi\theta\\ y = r\sin\pi\theta = 1.5\sin\pi\theta + 0.5\sin\theta\sin 20\pi\theta$

Answer 3

$$\begin{aligned} x(u) &= \left(\frac{3}{2} + \frac{1}{2}\sin 20\pi u\right)\cos\pi u \\ y(u) &= \left(\frac{3}{2} + \frac{1}{2}\sin 20\pi u\right)\sin \pi u \end{aligned}$$

$$0\le u \le 1$$

Mathematica code:

x[u_] = (3/2 + 1/2 Sin[20 \[Pi] u]) Cos[\[Pi] u];
y[u_] = (3/2 + 1/2 Sin[20 \[Pi] u]) Sin[\[Pi] u];
p = ParametricPlot[{x[u], y[u]}, {u, 0, 1}, Frame -> True, 
    Axes -> False];
Show[p, Graphics[{Gray, Circle[{0, 0}, 1]}], 
Graphics[{Gray, Circle[{0, 0}, 3/2]}], 
Graphics[{Gray, Circle[{0, 0}, 2]}]]