Question

Find the range of the following functions:
$f\left(x\right)=3\sin x+4\cos x-5$

Answer 1

Let $g\left(x\right)=3\sin x+4\cos x$

For $h\left(x\right)=a\cos x+b\sin x$ range varies from

$[-\sqrt{a^2+b^2}$$,\sqrt{a^2+b^2}]$

So, for $g\left(x\right)$ range is $[-\sqrt{3^2+4^2}$$,\sqrt{3^2+5^2}]=[-5,5]$

Thus for $f\left(x\right)=g\left(x\right)+5$, range is $[0,10]$