Question

Find the range of function $f\left(x\right)=\frac{\sin 3x}{\sin x}$

Answer 1

$f\left(x\right)=\frac{sin3x}{sinx}$

$\Rightarrow \frac{3sinx-4si{n}^{3}x}{sinx}$

$\Rightarrow \frac{sinx(3-4si{n}^{2}x)}{sinx}$

$\Rightarrow 3-4si{n}^{2}x$

Now, find the range of $3-4si{n}^{2}x$

we know, $0⩽si{n}^{2}x⩽1$

so, $0⩽4si{n}^{2}x⩽4$

$-4⩽-4si{n}^{2}x⩽0$

$3-4⩽3-4si{n}^{2}x⩽0+3$

$-1⩽3-4si{n}^{2}x⩽3$

So Range is $[-1,3]$