Why is $\sqrt{\sin^2 x} = |\sin x|$?

Question

Please thoroughly explain this as I am completely lost. I kind if understand but if $\sin x$ is negative, then $\sin^2 x$ is a positive number and then the square root would be $|\sin x|$for sure but wouldn't it also be $-\sin x$?

Answer 1

The absolute value is (sometimes) defined as $$ |x|=\begin{cases} x & \text{if $x\geq 0$}\\ -x & \text{if $x<0$} \end{cases} $$ So, $$ |\sin x|=\begin{cases} \sin x & \text{if $\sin x\geq 0$}\\ -\sin x & \text{if $\sin x<0$} \end{cases} $$ So, when $\sin x<0$, $|\sin x|$ means $-\sin x$. But, when $\sin x \geq 0$, $|\sin x|$ means $\sin x$.