Find the domain of a trig function

Question

Find the Domain of the function $y=\sqrt{1+2 \sin x}$. My solution:

1. $1+2 \sin x \geq 0$ (since square-root of a negative number is not defined)
2. $2 \sin x \geq -1$
3. $\sin x \geq -1/2$
4. $x \ge -\frac{\pi}{6}$
5. $x \geq (2n-\frac16) \pi$;

but the answer is $\left[\left(2n-\frac16\right)\pi;\left(2n+\frac76\right)\pi\right]$ for $n\in\mathbb Z$.

If step 4 says that any number greater than or equal to $-\pi/6$ will satisfy the equation, then any number greater than or equal to $-\pi/6$ will part of domain, so domain should be $\left[-\pi/6, \infty\right)$

Answer 1

Well done until the step 4. The problem is that $\sin x$ is not a monotonic function of $x$, that's why it is important to know how to solve trigonometric inequalities. The usual method is to find which $x\in[0,2\pi)$ admit the inequality and then add $+2\pi k$ with $k\in\mathbb Z$.

E.g. in your case only $x\in [0,7\pi/6]\cup[11\pi/6,2\pi)$ admit the inequality $\sin x\geq \frac12$. That's why the answer is different with yours. In fact, answer I've given here is the same as in your book due to the arbitrary $k$ in $2\pi k$ which you add.

Answer 2

You made a mistake in solving $\sin x \ge -\frac{1}{2}$. Solving it in the domain $-\pi < x \le \pi$ yields two solutions:

One of them $ -\frac{\pi}{6} \le x \le \pi$ you got correctly, but there is another one.