Question

Range of $f\left(x\right)=5+a\sin x+b\cos x$

Answer 1

Let $y=asinx+bcosx+5$
$\Rightarrow (y-5)=asinx+bcosx$
Recall that for all real values of $x$,

$\Rightarrow -\sqrt{a^2+b^2}\le (asinx+bcosx)\le \sqrt{a^2+b^2}$ ......(1)
$\Rightarrow -\sqrt{a^2+b^2}\le (y-5)\le \sqrt{a^2+b^2}$

$\Rightarrow 5-\sqrt{a^2+b^2}\le \left(y\right)\le 5+\sqrt{a^2+b^2}$

$\Rightarrow 5-\sqrt{a^2+b^2}\le f\left(x\right)\le 5+\sqrt{a^2+b^2}$
Hence the range is $[5-\sqrt{a^2+b^2},5+\sqrt{a^2+b^2}].$