I am trying to solve the following equation algebraically:
\begin{align} x/4 + \sin 3x &= 2.65 \ \end{align}
This equation has multiple solutions, and all of the solutions lie in the range $0 \leq x \leq 6\pi$.
To find the minimum solution, I first rearranged the equation to get $\sin 3x = 2.65 - x/4$. Then, I used the inverse sine function to rewrite the equation as follows:
\begin{align} 3x &= \sin^{-1}(2.65 - x/4) \ \end{align}
I plugged in the smallest possible value of $x$, which is 0, and found that the equation did not have a solution. Therefore, the minimum solution must be greater than 0.
To find the number of solutions, I used the periodicity of the sine function. The sine function has a period of $2\pi$, so I divided the range $0 \leq x \leq 6\pi$ by $2\pi$ to get:
\begin{align} \frac{6\pi - 0}{2\pi} &= 3 \ \end{align}
This suggests that the equation $x/4 + \sin 3x = 2.65$ has 3 solutions in the range $0 \leq x \leq 6\pi$. However, I am not entirely confident in my solution, so I would appreciate any feedback or suggestions from others in the forum.
Is it possible to solve this equation algebraically, or is some other method required? Any help would be greatly appreciated.
Thank you in advance for your assistance!