If $\sin \theta=\frac{1}{2}\left(x+\frac{1}{x}\right)$, then $\sin 3 \theta+\frac{1}{2}\left(x^3+\frac{1}{x^3}\right)=$

Let $\mathrm{A} \equiv(0,0), \mathrm{B}(3,0), \mathrm{C}(0,-4)$ are vertices of $\triangle A B C$, then the co-ordinates of incentre of $\triangle \mathrm{ABC}$ is

The equations of the tangents to the circle $x^2+y^2=36$ which are perpendicular to the line $5 x+y-2=0$ are

$$ \begin{aligned} & f(x)=\left\{\begin{array}{ll} 3-x, & -1 \leqslant x<0 \\ 1+\frac{5 x}{3}, & -3 \leqslant x \leqslant 2 \end{array}\right. \text { and } \\ & g(x)=\left\{\begin{aligned} -x, & -2 \leqslant x \leqslant 3 \\ x, & 0 \leqslant x \leqslant 1 \end{aligned}\right. \end{aligned} $$

then range of (fog) $(x)$ is