Question4. Solve the equation $\sin (8 x)=\sin (7 x) \cos (x)$ for $x \in(0, \pi)$.
ANS:

Studdy Solution
Check for solutions within the interval $x \in (0, \pi)$:
1. For $x = \frac{\pi}{14} + \frac{n\pi}{7}$, check values of $n$ such that $0 < x < \pi$.
- $n = 0$: $x = \frac{\pi}{14}$ - $n = 1$: $x = \frac{3\pi}{14}$ - $n = 2$: $x = \frac{5\pi}{14}$ - $n = 3$: $x = \frac{9\pi}{14}$
2. For $x = n\pi$, only $n = 0$ and $n = 1$ are valid, but $x = 0$ and $x = \pi$ are not in the open interval $(0, \pi)$.
The solutions for $x$ in the interval $(0, \pi)$ are:
$$x = \frac{\pi}{14}, \frac{3\pi}{14}, \frac{5\pi}{14}, \frac{9\pi}{14}$$