Answer to Question #89145 in Trigonometry for Sophia

Question

The sensor on a music player is circular. It has a diameter of 8 cm with its centre axis 5 cm from the bottom of the player. If a person moves their finger around the outside of the sensor 1 cm every 1 second, what adjustments could be made to the cosine function to model the movement of the finger?

Answer 1

The player looks like this:

1) Find the angular frequency of the finger:

$$\omega=\frac{\Delta\phi}{\Delta t}=\frac{l/r}{\Delta t}=\frac{1\text{ cm}/4\text{ cm}}{1\text{ s}}=\frac{1}{4} \text{ s}^{-1}.$$

2) The axis is $h=5\text{ cm}$ above the bottom of the player, and finger's highest and lowest position relatively to the axis is $A=r=d/2=4\text{ cm}$, so its motion can be described as

$$f(t)=h+A\text{cos}(\omega t)=5+4\text{cos}\Big(\frac{t}{4}\Big).$$