Question

The sound wave of two music notes can be represented by the function y = 2sin2x and y = −sin x, where x represents the time since the note is played. At what times will the frequencies of the sound waves be the same in the interval [0°, 360°)?

Answer 1

ANSWER

x = 0°, 180°, 210°, 330°

EXPLANATION

We have to find x when both functions have the same value,

$2\sin ^2x=-\sin x$

Add sin(x) to both sides,

$2\sin ^2x+\sin x=0$

And we have a quadratic equation, where the variable is sin(x). Factor out 2 sin(x),

$2\sin x(\sin x+\frac{1}{2})=0$

The solutions are,

$\begin{cases}\sin x=0 \\ \\ \sin x=-\frac{1}{2}\end{cases}$

Now we have to find the values of x whose sine is 0 and the ones whose sine is -1/2,

$\begin{gathered} \sin x=0\Rightarrow x=0\degree;x=180\degree \\ \\ \sin x=-\frac{1}{2}\Rightarrow x=210\degree;x=330\degree \end{gathered}$

Hence, the values where they are the same are 0°, 180°, 210° and 330°