5. Find the set of values of the expression $\frac{a \cos x - b \sin x + 2c}{\sqrt{a^{2} + b^{2} + c^{2}}}$ where $x, a, b, c$ are arbitrary numbers such that $a^{2} + b^{2} + c^{2} \neq 0$.
Solution: Let $\vec{n}=(a ; b ; c), \vec{m}=(\cos x ;-\sin x ; 2)$. Then $|\vec{n}|=\sqrt{a^{2}+b^{2}+c^{2}},|\vec{m}|=\sqrt{5}$. Consider the function $\quad f(x, a, b, c)=\frac{a \cos x-b \sin x+2 c}{\sqrt{a^{2}+b^{2}+c^{2}}}=\frac{\vec{n} \cdot \vec{m}}{|\vec{n}|}=|\vec{m}| \cos \alpha=\sqrt{5} \cos \alpha, \quad$ where $\quad \alpha-\quad$ is the angle between vectors $\vec{n}$ and $\vec{m}$. The set of values of the function $f$ is the interval $E=[-\sqrt{5} ; \sqrt{5}]$.

Answer: $E=[-\sqrt{5} ; \sqrt{5}]$.