Question

1. solve the equation on the interval $(-infty, infty)$ (find the general solution).

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

Explanation:

Step1: Isolate the sine term

Rearrange the equation: $\sin\frac{x}{2} = 1$

Step2: Find angle for sine=1

General solution for $\sin\theta=1$: $\theta = \frac{\pi}{2} + 2k\pi$, $k\in\mathbb{Z}$

Step3: Solve for x

Set $\theta=\frac{x}{2}$, so $\frac{x}{2} = \frac{\pi}{2} + 2k\pi$. Multiply by 2: $x = \pi + 4k\pi$

Answer:

$x = \pi + 4k\pi$ for any integer $k$