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

x = 0°, 180°, 210°, 330°

Step-by-step 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+(1)/(2))=0`

The solutions are,

`\begin{cases}\sin x=0 \\ \\ \sin x=-(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=-(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°