Question

Find the domain of definition of the following function:
$y=\log \sin (x-3)-\sqrt{16-x^2}.$

Answer 1

We know $\sin \theta \in (-1,1)$

For

$\log \left(\sin \right(x-3\left)\right)0<\sin (x-3)\le 10<x-3\le \frac{\pi }{2}3<x<3+\frac{\pi }{2}$

and

$\sqrt{16-x^2}16-x^2\ge 0(x^2-16)\le 0(x+4)(x-4)\le 0x\in [-4,4]$

Taking intersection, we get

$x\in (3,4]$