Question

1. a guitar string is about 50 cm long, measured to the point of strumming. playing middle c at forte volume causes it to move back and forth, making 3° angles with its original position. how far did the guitarist move the string sideways to achieve this note and volume?

Explanation:

Step1: Identify the right - angled triangle relationship

We can consider a right - angled triangle where the length of the guitar string is the hypotenuse ($l = 50$ cm) and the sideways displacement of the string is the side opposite the given angle ($\theta=3^{\circ}$). We use the sine function.
The formula for the sine of an angle in a right - angled triangle is $\sin\theta=\frac{opposite}{hypotenuse}$.

Step2: Solve for the opposite side

We know that $\sin\theta=\frac{d}{l}$, where $d$ is the sideways displacement, $\theta = 3^{\circ}$, and $l = 50$ cm.
So, $d=l\times\sin\theta$.
Substitute $\theta = 3^{\circ}$ (in radians, $\theta=3\times\frac{\pi}{180}=\frac{\pi}{60}$ radians) and $l = 50$ cm into the formula.
$d = 50\times\sin(3^{\circ})$.
Since $\sin(3^{\circ})\approx0.0523$, then $d=50\times0.0523 = 2.615$ cm.

Answer:

$2.615$ cm